The earth; its shape, size, weight and spin

THE

EARTH

J. H. POYNTING

CAMBRIDGE UNIVERSITY PRESS

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All rights reserved

THE EARTH

ITS SHAPE, SIZE, WEIGHT
AND SPIN

J. H. .JPOYNTING,

Sc.D., F.R.S.

Late Fellow of Trinity College

Cambridge ; Mason Professor

of Physics in the University

of Birmingham

Cambridge :
at the University Press

New York :
G. P. Putnam's Sons

1913

Camfaritige :

PRINTED BY JOHN CLAY, M.A.
AT THE UNIVERSITY PRESS

. With. the., exception of the coat of arms at
the fort, f'ir d^'igr* on the title page Is a
reproduction of one used by the earliest known
Cambridge printer, John Siberch, 1521

of.

PREFACE

rTlHE aim of this book is to explain in a general
-^- way, without mathematical detail, how the
shape and size of the Earth have been determined,
how its mass has been measured, and how we know
that it rotates, and so uniformly that it is a nearly
perfect time-keeper. Some account is given of the
tidal action which must gradually be reducing the
spin, a subject of which our knowledge is chiefly due
to the researches of Sir George Darwin.

Readers who wish to study further the matters
dealt with here, will find more detailed treatment
under various headings in the eleventh edition of
the Encyclopaedia Britannica. A bibliography of
each subject is there given.

On the Tides, Sir George Darwin's general account
should be read in his work on The Tides and
Kindred Phenomena of the Solar System.

J. H. P.

November, 1912.

267435

CONTENTS

CHAP.

PAGE

T m

I. THE SHAPE AND SIZE OP THE EARTH . . . i

II. WEIGHING THE EARTH

III. THE EARTH AS A CLOCK .

139

CHAPTER I

THE SHAPE AND SIZE OF THE EARTH

IF we stand on a hill top on a clear day, and look
over the lowlands stretching away from below, there
is nothing in what we see to suggest that we are
on the surface of a globe. There is no appearance
that the surface bends down from us as it recedes.
Rather does it seem as if the Earth slopes up towards
the horizon and as if the hill rises up in the middle
of a shallow cup.

When men first began to think about such obser-
vations as this, and to consider the shape of the
Earth, there was no obvious suggestion that they
were on a globe, and, naturally perhaps, the first
idea was that the Earth is a flat plain on which the
mountains are creases, a flat ' firmament in the midst
of the waters.'

Gradually, however, observations accumulated
which could not be reconciled with the flatness of
the Earth. A traveller, journeying from a mountain
range, found on looking back that the mountains
not only grew smaller and smaller but that they

2 , THE EARTH [OH.

sank and at last dropped down altogether out of
sight.

When men began to go down to the sea in ships
and ventured far out on the waters, the new land to
which they sailed appeared first as one little peak,
then as a range, and at last the whole land stood
above the water. These observations were difficult
to reconcile with the idea of a flat Earth, but easy to
explain if it was round.

The doctrine of the roundness of the Earth, then,
gradually replaced the doctrine of its flatness. But
there was a long fight of nearly 2000 years between
the doctrines. When Columbus at the end of the
15th century proposed to reach India by sailing to
the west instead of to the east, arguing that as the
Earth was round, the other side might be reached
either way, his opponents, holding that the Earth
was flat, regarded him as a fool and a heretic. It
was urged that if the Earth were round, men on the
opposite side would be walking with their heels up-
wards, that the trees would be growing with their
branches downwards, and that it would rain, hail and
snow upwards. All this appeared to them absurd,
for they did not realise that the tendency to fall is a
tendency to fall towards the centre of the Earth.
They thought of ' falling' as a motion in the same
direction everywhere, and anything loose on the
other side of the Earth, if that other side could be

i] SHAPE AND SIZE OF THE EARTH 3

conceived as existing, should fall away from the sur-
face. They argued that, in order to travel from that
other side to this, a ship would have to climb up the
sea as if it were climbing up a mountain slope, and
that no wind would suffice to drive it up. It was
even urged that the roundness of the Earth was
inconsistent with the resurrection of the body. For
the dead on the other side of the globe would rise on
this side with their heels uppermost.

Columbus fought the last fight against a flat
Earth, and won. He sailed to the west and found,
not indeed the India which he had hoped for, but the
West Indies. Soon after, the journey round the
world was made, and the Earth was henceforth a
globe for all who could study the evidence. Let
us consider this evidence in its most conclusive
form.

If we watch the stars, by night, at a place in
this part of the world, we see that one star, the
pole star, does not noticeably change its position,
and that all the other stars circle round it. When
we make careful measurements we find that the
pole star is not quite fixed but goes in a small circle
round a centre, which we may conveniently call the
sky pole, and it is this sky pole round which all the
other stars circle.

If we travel northwards, the stars still circle
round the same pole, but the pole itself rises higher

1—2

4 THE EARTH [CH.

in the sky. The fundamental fact is, that for the
same distance of travel due north, the pole rises the
same number of degrees, wherever our starting-point
may be.

If we are at sea where the horizon is definite we
may measure the height in degrees above that horizon.
If we are on land where the horizon is conditioned by
the elevation of the land and is therefore not a
definite line, we may measure the distance in degrees
from the zenith, the point directly overhead ; and the
zenith distance is 90° minus the distance from the true
horizon, where, by the true horizon, we mean that
which the sea-line would give if we had sea in place
of land. Thus at a point near Nottingham the sky
pole is 53° above the true horizon or 37° below the
zenith. If we travel due north 69 miles, to a point
near York, the sky pole is 54° above the true horizon
or 36° from the zenith. Or if we go across to Ireland,
at Cavan it stands 54° above the true horizon, while
69 miles due north at Londonderry it stands 55°
above the horizon. Or taking a longer distance, at
Coventry it stands about 52 J° above the horizon,
while at Sand wick in the Shetlands 552 miles due
north it stands about 60|° above the horizon, having
risen 8° for a travel 8 x 69 miles northward. Every-
where the rise is very nearly at the same rate of 1°
for 69 miles' travel northwards, not exactly the
same, as we shall see later, for the distance increase8

i] SHAPE AND SIZE OF THE EARTH 5

slightly as we go from the equator towards the pole,
but the increase is very slight.

Postponing for the present the description of
the way in which pole height and distances are
measured we may see at once that the relation we
have stated is quite inconsistent with a flat Earth.
Let us take the last case of Coventry and Sandwick
and for simplicity of statement let us think of the
pole star as actually at the sky pole. It need not
affect the conclusion, for at one point in its circle
the pole star will be at the same height above the
horizon as the sky pole, and we may choose that point
for consideration.

Let C (fig. 1) represent Coventry on a flat Earth,
and $ Sandwick 552 miles due north. At C make
the angle PON = 52£°
and at S make the angle
PSN = 60¥, the two
lines CP and SP meet-
ing in P the pole star.
It is easy to calculate
by trigonometry, or to
find by direct measure- L K H c sf

ment of a carefully Flg< lm

drawn figure in which OS represents 552, that to
the same scale PN is about 2740 and CP is about
3450. That is, a flat Earth requires that the pole
star is less than 3500 miles from Coventry, an utterly

6

THE EARTH

[CH.

absurd result as we shall presently see. But passing
this by, let us mark points H, K, L, distant 552
miles, 2 x 552 miles, and 3 x 552 miles respectively
from C. On a flat Earth the angles PON, PHN,
PKN, PLN do not decrease successively by equal
amounts. Careful measurements on a large figure
suffice to show this. On the real Earth they de-
crease successively by 8°, so that the Earth cannot
possibly be flat.

Another set of measurements would show equally
well that the Earth is not flat, and they are worthy
of description inasmuch as they give us conclusive
evidence of the true form.

If the Earth were flat and the pole star were
vertically above a point O (fig. 2 (a)) on the flat sur-

Fig. 2 (a).

Fig. 2 (b).

face, two lines of travel SN, S'N', each towards due
north, would be straight lines and would meet at

i] SHAPE AND SIZE OF THE EARTH 7

0. The distances between the two lines at SS' or
NN' would be proportional to the distance along
either from O. Two travellers along these lines
would approach each other by equal amounts for
equal distances travelled northward. But the law of
approach is quite different. If we start from two
points on the equator, the line on the Earth's surface
for which the sky pole is on the horizon, and travel
due north from each, that is, always towards the
point on the horizon immediately under the sky
pole, the distance between the two lines of travel,
measured along a line of equal pole height, is pro-
portional to the cosine of the angle through which
the pole has risen in the sky. Or if we draw a
quarter circle (fig. 2 (6)), and represent the distance
travelled along each line by the length SN along
this circle, the angle SON being the rise of the sky
pole, then the distance between the lines, if measured
along a line of equal pole height, will be proportional
to^Of.

I have thought it worth while, even though the
doctrine of a flat Earth has long been abandoned,
to examine carefully the evidence which led to its
abandonment. For that examination enables us to
see that our ancestors were not so wrongheaded in
holding the doctrine as, at first thought, they might
seem to have been. They accepted the most obvious
account of appearances, a flat Earth. They observed

8 THE EARTH [CH.

that everything tended to fall straight down to the
Earth ; everywhere, as far as they could tell, in the
same direction ; or down-ness was universal. Having
once taken this view, it was a real difficulty, rightly
felt, that bodies could remain on the surface at the
Antipodes. They should fall down into space there
just as they fall down to the surface here. They had
no measurements contradicting their view, nor had
they means to make the measurements had they
wished to do so.

Another piece of evidence, commonplace to us,
was utterly closed to them. They had no difficulty
in thinking of the Sun as rising up over the edge of
the Earth and illuminating the whole surface at
once. It was only with the invention of trustworthy
portable clocks that it could be clearly proved that
sunrise, noon, and sunset take place earlier and earlier
the further we go eastward, later and later the
further we go westward. Every traveller across the
Atlantic knows that the clocks on board have to
be altered each night to make them agree even fairly
with the Sun, and every follower of cricket knows that
the Australians may have finished a match before we
breakfast.

We shall now examine the interpretation which
we are obliged to give to the two sets of measure-
ments which we have described, viz.:

1. That the sky pole rises through equal angles

i] SHAPE AND SIZE OF THE EARTH 9

towards the zenith for equal distances travelled due
northwards.

2. That two lines on the Earth's surface, each
drawn due northwards, are a distance apart, if
measured along a line of equal pole height, pro-
portional to the cosine of the pole height or pro-
portional to the length NM in fig. 2(6).

It is first necessary to show that we can fix a
definite direction in space, wherever we may be on
the Earth's surface, by drawing a line to one of the
fixed stars.

Wherever we may be on the Earth, if we see the
same groups of stars those groups form the same
patterns in the sky. This shows at once that they
are vast distances away. For if we look at any
arrangement of objects, the less does the arrange-
ment appear to change with a change in our position
the further the objects are from us. As we walk
along a road, the view of a house by the roadside
changes almost with every step. A wood further
back alters more slowly. Still, as we move the trees
do appear to change places, a nearer tree being now
in front of one, now in front of another of those
further back. But a range of distant hills may show
just the same appearance even though we move
hundreds of yards along the road. No measurements
which have been made, even with the most powerful
telescopes, from different parts of the Earth's surface

10 THE EARTH [CH.

at the same time have ever shown the least differ-
ence in pattern of the stars in any constellation,
and we are forced to conclude that the stars are
immensely distant in comparison with any distance
we can set out on the Earth. Indeed, the pattern
only changes very minutely if we use the vastly
greater distance afforded by the motion of the Earth
round the Sun from one side of its orbit to the other.

It follows that the direction of any one fixed star
is, as nearly as we can measure, the same as seen
from all parts of the Earth, or that straight lines
drawn from all points to the star are parallel. This
will hold good if, instead of any particular star, we
take the point about which the stars in their patterns
appear to circle ; that is to say, the sky pole.

Let $ (fig. 3) be a point from which the sky pole
is seen along the line SP, and let N be a point due
north of S from which the pole is seen along the line
NPf parallel to SP. If SZ and NZ' are the verticals
at S and N, that is, the lines directed towards their
respective zeniths, the angle ZSP is greater than the
angle Z'NP', and, as we have seen, if SN is 69 miles
it is greater by 1°. The surface therefore bends
away from a fixed direction, that of the sky pole, by
equal amounts in equal distances. This shows at
once that in going northwards we are travelling in a
circle, for that is the only curve which bends through
equal angles in equal distances. If we produce the

i] SHAPE AND SIZE OF THE EARTH 11

two verticals ZS, Z'N to meet in 0, 0 is the centre
of the circle. If we produce P'N to meet OS at ~R,
the angle

NOS = NES - ONE = PSZ - P'NZ'.

If then SN- 69 miles, NO 8— 1°, and since there are
360° in the complete circle its circumference is
69 x 360 = 24,840 miles,

and the radius is 3950
miles, since the circum-
ference of a circle is
6*283 x radius, very nearly.
These numbers are not
quite exact, since the dis-
tance 69 miles for 1° rise
is not quite exact.

We have supposed that
we are travelling north-
wards where the northern
sky pole is visible. But
if we travel southwards beyond the equator, where
the southern sky pole is visible, we have the same
rise of 1° per 69 miles travel, so that we move in a
circle of the same size everywhere.

The surface of the Earth must therefore have a
shape such that a plane drawn through the vertical
at any point, and through the line to the sky pole,
must cut the surface in a circle with radius about

Fig. 3.

12 THE EARTH [OH.

3950 miles. There are three and only three shapes
for which this could be true, a cylinder, like a round
ruler, an anchor ring, and a sphere. The second set of
measurements on p. 9, giving the law of approach
of two lines both running due north, at once enables
us to decide between the three. On a cylinder the
two lines would always be the same distance apart.
On an anchor ring they would approach, but more
slowly than the measured rate. On a sphere alone
would they approach at the measured rate. We may
easily see that a sphere gives this rate. For if PP'
(fig. 4) is the diameter of the sphere parallel to the
direction of the sky pole, and if two planes through
PP' cut the surface in the circles PNEP' and
PN'E'P', let

PN=PNf,

and let MN, MN' be perpendiculars to PP'. Let
NN' be an arc of a circle with centre M. If a is the
angle between the two planes measured in radians,

or NN' is proportional to the length NM. If we
denote by X the angle NOE, where E is halfway
between P and P', then

Hence NN' is proportional to cosX, and X is
easily seen to be the height of the sky pole above
the horizon.

i] SHAPE AND SIZE OF THE EARTH 13

Our measurements, then, lead us to the irre-
sistible conclusion that the Earth is, at least very
nearly, a round globe, with a radius about 4000 miles.
If we draw circles round it passing through P and
P', and running due north and south, they are lines
of longitude. If we draw circles round it with their
centres at different points in PP',
they are lines of latitude. In fig. 4
the angle NOP is the angle between
the vertical at N and the direction
to the sky pole. The angle

is the angle which the sky pole makes Ej
with the horizon at N, and since
NOE is termed the latitude of N,
the height of the sky pole above the
horizon at a place is equal to the
latitude of that place.

The determination of the size of
the globe depends on the measure-
ment of the angle which the sky pole
makes with the horizon or with the zenith, and on
the change in this angle when we travel measured
distances north or south. We must now consider
how we can assert that, for instance, the pole stands
52^° above the horizon at Coventry and 60^° above it
at Sandwick, and how we can measure the distance

14

THE EARTH

[CH.

between these two stations and assert that it is
552 miles.

First, as to the measurement of the pole height.
We may suppose that for this purpose we use a
theodolite, an instrument which is represented
diagrammatically in fig. 5. BJ5 is a tripod base on

H/

f\

A

I I

" I

Fig. 5.

levelling screws. Only two feet of the three are
shown. On the base is fixed a horizontal circular
plate HH divided to degrees and fractions. Above
this is a framework essentially consisting of two
pillars, of which the front one only, P, is shown.
This framework is mounted on an axis which fits

i] SHAPE AND SIZE OF THE EARTH 15

a vertical bearing in the tripod base, and attached
to it are two arms aa with verniers on them marking
the position of the framework on the horizontal
circle HH. At the tops of the pillars are two V
bearings for a horizontal axis A, which carries a
telescope TT. On this telescope is fixed a vertical
circle VV divided to degrees and fractions. Two
arms bb with verniers on them are fixed to the pillar
P, and as the telescope revolves, and carries the
circle VV with it, these arms mark the angle on the
circle through which the telescope has revolved. In
the telescope are two cross wires or some equivalent
arrangement, so that the image of the object looked
at may be brought exactly to the same point, always
in the middle of the field of view. We need not
enter into the modes of adjusting the two axes so as
to be respectively vertical and horizontal. These
will be found in any book on surveying.

Now suppose that the telescope is directed to the
pole star, and that its position on the vertical circle
is read. If it can be then turned round exactly into
the horizontal direction and its position again be
read the difference will give the height of the pole
star above the horizon. But it is only at sea, by day,
that we have a definite horizon to turn to, and even
then allowance must be made for the fact that the
line from a point any distance above sea-level slopes
downwards to the horizon, owing to the curvature of

16 THE EARTH [CH.

the Earth. We can, however, entirely dispense with
the horizon by using a horizontal reflecting surface
such as is afforded by a small trough of mercury.
One way of using the mercury-trough consists in
placing it between the pillars of the theodolite, and
pointing the telescope vertically down towards it.
It is known when the telescope is exactly vertical by
observing when the reflexion in the mercury of the
cross wires in the eye-piece coincides with the actual
cross wires, a special illuminating device, which we
need not describe, being used to make the cross
wires and their reflexion visible. The position of the
telescope is then read on the vertical circle, and
when it is turned through 90° from this position it is
pointing to the true horizon.

But the pole star is not exactly at the pole of the
sky. It circles round it. If, however, we measure
its height when at its highest point above the horizon
and its height when at its lowest point, and take
the mean of these, we get the height of the centre
round which it is circling.

We have taken the pole star as an example of the
method of determining the pole height. Any one of
a large number of stars would serve equally well, for
their angular distance from the pole is accurately
known from measurements which have been made at
observatories, such as that at Greenwich. If then we
measure the height of one of these stars when it is

i] SHAPE AND SIZE OF THE EARTH 17

crossing the meridian, and therefore at its highest or
lowest point in the sky, it will easily be seen that we
may deduce the height of the pole.

There is a correction to be made to the observed
height of a star owing to the fact that light does
not come straight through the atmosphere unless it
comes from the zenith, but bends down somewhat.
The direction in which a star appears to be is the
direction in which a ray from it enters the telescope.
The star therefore is not quite so high in the sky as
it appears to be. This displacement has been deter-
mined by finding what correction must be made to
the observed heights of a star as it circles round the
pole to make them all fall on the same circle, and
tables are made giving the correction to be applied
to the observed height to turn it into the true height
for every position of a star.

Now as to measurement of distances on the Earth's
surface. How is the size of a county, a kingdom or
a continent determined ? We might chain lengths as
a surveyor chains a small plot of land, but the labour
for any great distance would be immense and the
undulations of the ground would bring in errors of
very considerable amount.

Fortunately there is a method which enables us
to measure distances from one point to another
hundreds of miles apart with an error hardly more
than a few feet. This is the ' base-line' method,

P. 2

18 THE EARTH [OH.

and it depends upon the fact that in all triangles
with the same three angles, the sides are in the same
proportion to each other, so that if in any one
triangle we know the length of one side which we will
take as the base, and if we know the number of
degrees in each of the two angles at the base, we can
calculate the lengths of the other two sides of the
triangle by known rules of trigonometry without
further measurement.

A

B A b

Fig. 6.

We may illustrate the principle of the method by
a very simple case. Suppose that we wish to measure
the distance between two points A and C (fig. 6), say
on opposite sides of a river, without crossing the
river. Let A be on the observer's side. The observer
is to cut a triangle out of cardboard abc. He must
fix the corner a at A, so that looking along ac he
sees Ct while on looking along ab he has a straight
course which he can traverse towards D. Having

i] SHAPE AND SIZE OF THE EARTH 19

marked the line AD he moves along it, always
keeping the base ab of the card in the line AD
until he finds that the corner b has come to a
point B, such that looking along be he sights C
again. It is obvious that the triangle BA C is similar
to the triangle bac, so that AC bears to AB the
same ratio that ac bears to ab. We have therefore

If then the observer measures ac, ab and the base
AB, he can at once calculate the distance AC.

We can easily see that if we make a mistake in
the angle at B it leads to a much more serious error
in the measurement of AC, when the base line AB is
small, than when it is not very different in length
from A C. Let us take the two cases represented in
fig. 7, (a) with a base line comparable with AC,
(b) with a very short base line, and let us suppose
that we do not move quite to the right point B but
go by mistake only to B', making an error in the
angle equal to BCB', about the same in each case, so
that BE' will not be very different in the two. The
error in the value of A C will be

j
ab AB

Since -j-^ is much greater in fig. 7(b) than it is in

2—2

20 THE EARTH [OH.

fig. 7 (a), the error is obviously greater in the former.
Hence the base should not be very small compared
with the distance to be measured if accurate measure-
ment is to be made.

B B ABBA

Fig. 7 (a). Fig. 7(6).

In applying the principle to Earth measurements,
that is, to measurements such as are made in our
Ordnance Survey, the first step is to choose a level
surface on which a straight course may be traversed
between two points A and B several miles apart.
The distance AB constitutes the 'base-line' and it
is measured as accurately as possible. Every sub-
sequent measurement of distance depends on this
first measurement. We know the size of a country
the size of the Earth, the width of its orbit round
the Sun, the distances of the planets and the fixed
stars in terms of a base line on the surface of the
Earth.

I] SHAPE AND SIZE OF THE EARTH 21

In one important respect, to be described below,
the method of measuring a base-line has been
changed lately. But the method is still the same
in principle as the older methods, and we shall
describe these, as they bring into prominence the
difficulties to be overcome. If the oldest method is
to be followed, two or more rods, each several yards
or metres long, and usually of metal, are employed.
We will suppose that we have two of these. The
exact length of each rod is determined beforehand
by comparison with a standard yard or metre in a

da' b'

Fig. 8.

laboratory. One rod db (fig. 8) is placed on sup-
ports with its end a at the end A of the base line
and its length is adjusted as exactly as possible in
the line AB. The other rod, cd, is then supported
in the line of continuation of ab with a small gap
between b and c, so that there shall be no risk of
displacement of ab by contact with cd. The width of
the gap has been measured in various ways. In a
way once used a little graduated wedge was dropped
into it with the narrow end downwards and the
depth of its descent gave the width of the gap.

22 THE EARTH [CH.

Another way is applicable to bars in which the
length used is not that between the two ends of the
bar but that between two marks on its upper surface.
Two microscopes are fastened together with their
axes a known distance apart. One of these sights
the mark on ah near b, the other sights the mark on
cd near c, and if the marks do not appear exactly in
the middle of the fields of view allowance can be
made. The gap or interval having been measured,
ab is taken up and then put down on supports in the
position a'b' beyond and in a line with cd. The gap
da' is measured and then cd is taken up and put
down beyond a'b', and so on, until B is reached.
A and B may be marked by fine lines ruled on metal
plates and the length of AB is the sum of the
lengths occupied by the bars in all their positions
plus the sum of the widths of the gaps.

Inasmuch as metal bars in general expand with
rise of temperature, each bar or rod used in the
older measurements had its length determined at
some standard temperature and the change in length
with any change in temperature was also measured.
When a bar was being used elaborate precautions
were taken to ward off inequalities of temperature
in different parts of the bar and great changes of
temperature from the mean, the bar being usually
contained in a long double box open at the ends.

Later, Colonel Colby devised a measuring rod

i] SHAPE AND SIZE OF THE EARTH 23

for the Indian Survey which was compensated for
temperature changes on a principle first used for
pendulums by Ellicott.

Fig. 9 represents the skeleton of the apparatus.
Two bars of different metals are used, one with con-
siderably greater expansion than the other. For
simplicity, let us suppose their expansions per degree
rise to be as 3 : 2, about the ratio for brass and iron.
AB is the more and CD the less expansible, and

1
1
/
/

I

C N D

\

\

\
\
\
\

-\D'

1
1

1

1

1

A' A M B B'

Fig. 9.

they are firmly fixed at their middle points to a cross
bar MN. At their ends are rods ACE, BDF jointed
or pivoted at A C, BD.

Let the apparatus at first be as indicated by the
continuous lines in the figure. Then let expansion
take place as indicated by the dotted lines, to A'B',
C'D'.

Since A A = 1(7(7', AC' prolonged will cut A C in
E where AE = §CE or AE = 3 AC, i.e. E is at a fixed
distance along AE. So too F is at a fixed distance

24 THE EARTH [CH.

along BF and the distance EF remains unchanged
by the expansion.

Such a compound rod requires, for successful
action, that the two bars shall have the same tem-
peratures, or at least the same temperatures at equal
distances from MN. In practice these rods have
hardly borne out the expectations formed for them
at the beginning.

Now there appears to be a prospect that com-
pensated measuring rods, like compensated pendulum
rods, will be entirely superseded by wires or tapes
of ' invar/ an alloy of nickel and steel discovered by
M. Guillaume, which hardly changes its dimensions
with ordinary changes of temperature. The wires or
tapes are very much longer than the rods — as much
as 100 feet — and each is stretched when in position
by a definite and constant pull. Small variations of
temperature from point to point are unimportant.
The process of measurement becomes less cumbrous
and the time required is much less1.

Base-lines several miles in length are measured so
accurately, either by the older or newer methods,
that several repetitions of the measurement agree
together within a fraction of an inch.

When the length of the base line AB (fig. 10) has

1 An account of the measurement of a Geodetic Base Line at
Lossiemouth, in 1909. Ordnance Survey Professional Papers, New
Series, No. 1.

i] SHAPE AND SIZE OF THE EARTH

25

been determined the next step is to use it as the
base of a triangle ACB, of which the vertex C is
some distant but easily seen point, such, for instance,
as a mark on a staff on the top of a church tower.
A theodolite, the instrument represented in fig. 5,
is placed at A and when the telescope sights B, the
position on the horizontal circle is read. Then the
telescope is moved round to sight C and the hori-
zontal circle is read again.
Thus the angle CAB is
known, as it is the angle
through which the telescope
is turned. Then the same
process is carried out at B
and the angle CBA is known.
Then having the length of
AB and the angles at its
ends we can, by the aid of
trigonometry, calculate the
lengths of A C and BC.

Either of these lines may be used as a base for a
new triangle. For instance, BC may be used in a
triangle CDB where D is perhaps a staff on a pile of
stones on a hill top. The theodolite is used at B
and C to measure the angles CBD and BCD. Then
BD and CD can be calculated in terms of BC, and
as this is known they too are known. Either of these
may be used as a base for a new triangle, CD for

Fig. 10.

26 THE EARTH [CH.

instance, carrying us to a new point E. So gradually
a whole country or even a whole continent may be
covered with a network of triangles, and all the
sides of all the triangles are found in terms of the
base-line. This process is known as triangulation
and, when it has been carried out, it is only a matter
of trigonometrical calculation to determine the dis-
tance between any two points in the network, however
far apart.

For simplicity the method has been described as
if all the points lay on a flat plain. But in reality
the measurements are not quite so simple as if this
were the case. Thus if in the triangle BCD, CB
and CD are not horizontal we do not measure
exactly the angle BCD. To do that the axis of the
theodolite would have to be tilted slightly so as to
be perpendicular to the plane CBD, an adjustment
which could not be made accurately even if it were
desirable. But we can adjust the axis accurately in
the vertical at C. In turning the telescope round
a vertical axis from sighting B to sighting Z>, we
really measure the angle between the vertical planes
through CB and CD, not quite the same thing
as the angle BCD. To find the latter we also have
to observe on the vertical circle how much the
telescope is tilted from the horizontal. There are
rules which would enable us, from these measure-
ments, to determine the angles in the triangles and

i] SHAPE AND SIZE OF THE EARTH 27

the length of side. But the plan actually followed
consists in projecting the straight line triangles down
on to the curved surface which the ocean would
give if there, that is, to the sea-level surface.

The observations enable this to be done and the
network of actual triangles is replaced by a network
of spherical triangles bent so at to fit the surface at
sea level.

Even in getting the directions of the straight lines
between the stations there is another troublesome
correction to be made. A ray of light only passes
straight through the air when it comes from over-
head. In all other cases it is curved and therefore
an object appears in a slightly different direction
from that in which it would be if the air were removed.
This effect of the air, the error of refraction, has been
studied and can be allowed for.

The Ordnance Survey in this country began in
1784, with the measurement of a base-line on
Hounslow Heath about five miles long. The original
idea was to form from this base a network of triangles
over the southern counties to the neighbourhood of
Dover, whence it could be carried across the Channel
to France. There a similar network was being
formed and when the two were connected so as to
form one system the difference in longitude between
Greenwich and Paris, the ultimate aim, could be
determined. This was soon effected, but fortunately

28 THE EARTH [OH.

the work did not stop here. The government de-
cided to continue the triangulation over the whole of
the British Isles, and so began the great survey of
the kingdom which was only completed in 1852. In
its course other base-lines were measured, as for
instance one on Salisbury Plain nearly seven miles
long, and one on the shores of Lough Foyle nearly
eight miles long. Triangles were formed from the
Welsh and Scotch mountain tops to the tops of Irish
mountains, and from the north of Scotland and
Orkney to Fair Island and Foula and so on to
Shetland, and so one triangulation embraced the
whole kingdom.

As a test of the accuracy of the work a series of
triangles was selected, starting from the Lough Foyle
base and ending in a triangle of which the Salisbury
Plain base formed one side. The length of the latter
base could then be calculated from the measurement
of the former and the measurements of all the angles
in the intervening triangles. The calculation differed
from the actual measurement by less than 5 inches.
With other pairs of bases the same kind of agree-
ment was obtained. All the lengths calculated in all
the triangles are therefore in all probability more
accurate than 1 in 10,000.

As we have seen, the original aim was to connect
up with a continental survey. This connection has
been repeated, and our triangulation now forms part

SHAPE AND SIZE OF THE EARTH 29

of a network covering all Europe. India and South
Africa have triangulations which will extend, and
at no distant date one system will no doubt spread
over the three continents of the Eastern hemisphere.
Another triangulation will cover the western con-
tinents, and distances will be known between points
separated by nearly half the Earth's circumference.
Meanwhile it suffices for the purpose of determining
the size of the globe that we are enabled to find the
exact distance between such points as Sandwick and
Coventry with a difference of pole height of 8°.

So far we have only considered the measurements
as showing that the Earth is round. It is very nearly
but not quite round. Sir
Isaac Newton showed that
if it were liquid, the spin
round its axis once in 24
hours should make it bulge
slightly at the equator and
draw in slightly at the
poles. A section through
the axis would therefore
not be a circle but an oval. In fig. 1 1 the departure
from the circular form is enormously exaggerated,
but the exaggeration enables us to see at once that
the curve bends round more in a given distance in
the equatorial regions EE than in the polar regions
PP. Or if the Earth has the shape which Newton

30 THE EARTH [CH.

assigned to it the vertical changes as we travel north
more rapidly in the neighbourhood of E than in the
neighbourhood of P, and the length of a degree
of latitude is less near the equator than near the
pole.

But some measurements made by Cassini early in
the 18th century appeared to show that the length
of a degree of latitude was less in the northern part
of France than in the southern part, and a school of
astronomers maintained that the earth was elongated
towards the poles — lemon-shaped instead of orange-
shaped.

It was difficult to resist the reasoning of Newton,
reasoning which would apply to a plastic solid earth,
as well as to a liquid earth, but Cassini's measurements
were against the result. To decide the question the
French Academicians sent out two expeditions, one
to Peru in 1735 and the other to Lapland in 1736, to
determine the length of a degree of latitude in each
region. The Peruvian expedition selected a district
at the equator near Quito, and the Lapland expedi-
tion a district near Tornea about 60° N. lat., which
was as near the pole as was convenient. In each
case a base-line was carefully measured (in Peru two
were measured, the second one for verification), and
from it a triangulation was carried out, so that the
length of a certain line running N. and S. was deter-
mined, in Peru about 200 miles, in Lapland over 60

i] SHAPE AND SIZE OF THE EARTH 31

miles. The change in the vertical between the ends
of these lines was measured by astronomical obser-
vations and the results were that near Quito

one degree of latitude = 56753 toises,
and near Tornea

one degree of latitude = 57438 toises,
a toise being about 6 feet.

Though there was some uncertainty about the
Lapland value there could not be any doubt that the
northern degree was the greater, and so it was de-
finitely decided that the figure of the Earth was
more nearly that predicted by Newton than that
which Cassini believed to be given by his measure-
ments.

There is an interesting story about the Peruvian
base-line near Quito. De la Condamine, a member of
the expedition, erected two small pyramids exactly
at the ends of the base-line, so that its position
should be permanently recorded. But soon after
his return to France he learned that the Spanish
government, probably in disapproval of the inscrip-
tions, had ordered the pyramids to be destroyed.
Subsequently orders were given for their re-erection.
Whymper in his Travels amongst the Great Andes
(p. 292) tells how he visited the re-erected pyramids
in 1880. 'The pyramid (of Oyambaro or Oyambarow)
which now approximately marks the southern end of
the base is about 1000 feet distant from the place

32 THE EARTH [OH.

where the stone reposes [the stone on which was the
offending inscription], situated in a field of maize,
and is neither the original pyramid nor the one
which was erected to replace it. I was informed on
the spot that it was put up about thirty years earlier
by a President of Ecuador, who so little appreciated
the purpose for which it was originally designed
that he moved it some hundreds of feet on one
side, in order, he said, that it might be better
seen.'

Subsequent measurements made in all parts of the
world do not exactly fit in with any simple mathe-
matical figure. A section through the axis of rotation
is nearly but not quite an ellipse, and a section through
the equator is nearly but not quite a circle. The
departures from these regular figures are, however,
very small — at the equator at sea-level not apparently
nearly so much as a mile. We shall be making only
a very minute error if we think of the section
through the polar axes as an ellipse with the polar
axis shorter than the equatorial axis by 1 in 293,
and the surface as having the form made by the
rotation of this ellipse round its shorter axis. If
there were open sea at the poles, the axis from sea
to sea would be, according to Col. Clarke (Geodesy,
p. 319), 7899J miles, while the equatorial diameter
from sea to sea is 7926 \ miles, probably within J mile
in each case.

I] SHAPE AND SIZE OF THE EARTH 33

The Earth, then, is very round. If an exact model
were made the size of a two-inch billiard ball, we
should just be able to see that it was flatter at the
poles, and, no doubt, in rolling it would exhibit its
want of roundness. The highest mountains would
be represented by elevations of ^J^ inch, say by the
thinnest smear of grease, the deepest oceans by the
spreading of a drop into a film but 7£o th inch thick.

To sum up, we find the size and shape of the
Earth by measurements of lengths on its surface,
starting from a base-line, and by measurements of
the angles which some of the fixed stars make with
the zenith when crossing the meridian. In making
the astronomical measurement, it is assumed that
the stars observed are so far off that lines drawn to a
given star at the same instant from different parts of
the Earth's surface may be regarded as parallel. This
is justified by the observation that the patterns made
by the constellations do not show any appreciable
change when looked at, at the same time, from places
as wide apart as we can have them, and by choosing
stations on opposite sides of the Earth we can have
them nearly 8000 miles one from the other. If further
justification were needed, it would be afforded by
the fact that the distances of many of the nearer
fixed stars have been measured, and that these dis-
tances are so enormously great compared with the
8000 miles diameter of the Earth that the want of

p.

34 THE EARTH [OH.

parallelism in lines to a star seen from opposite sides
of the Earth is utterly insignificant.

We shall conclude this chapter by a short account
of the way in which the distances of the nearer stars
are found. It is again a base-line method, but the
base is the diameter of the Earth's orbit and two
stations can therefore be used 180 million miles
apart. From stations so widely separated the pat-
tern of the constellations does change slightly in
some cases, the brighter and presumably nearer stars
shifting slightly on the background of the fainter and
presumably remoter stars as we move in the course
of six months from one end to the other of the vast
base-line stretching across the orbit.

Let us suppose that we have selected a star for
examination, and that near it, and seen at the same
time in the telescope, are faint stars which do not
change their relative positions and so are presumably
enormously distant. Let AB (fig. 12) be two positions
of the Earth on opposite sides of the sun S, six months
apart in time, 180 million miles in distance. Let C be
the star to be examined and, for simplicity, we will
suppose that CS is perpendicular to A SB. Of course
the figure enormously exaggerates the angle A CB.
It is far too minute with any actual star to be shown
in a figure. If, now, we can measure1 the angle A CB

1 It is usual to give in tables the value of ACS=\ACB and this is
termed the parallax of the star.

i] SHAPE AND SIZE OF THE EARTH 35

we can at once determine the distance AC or BC,
for it can be shown that

. ~ 38 million million miles

number of seconds of arc in ACS'

Let there be a star seen near C, and in the plane
of A CB, which shows no sign of change of position
with regard to its fainter
neighbours and let ADl, BDZ
be lines drawn to it, pre-
sumably parallel as far as any
possibility of observation goes,
and let CDS be a third parallel.

Since CD3 is parallel to
BD2 the angles D3CB and
CBDz are equal to each other.
And since CDS is parallel to
ADl the angles I)SCA and
CA Dl are equal to each other.
But

ACB = D,CB - D8CA

In one method there is, in the eye-piece of the
telescope used, measuring apparatus by which the
angle CAD± can be measured and then, some six
months later, the angle CBD2 ; or photographs may
be taken and the positions of the stars on these
may be measured with a microscope. But even with

3—2

36 THE EARTH [CH.

the nearest fixed star, a Centauri, a bright star in the
southern hemisphere, A CB is only 1J seconds. Very
minute errors in the measurement of CADl and <7J5Z>2
may produce very serious errors in the result, and it
is only comparatively lately that the measurements
made for the same stars by different observers have
shown good agreement.

The distance of a Centauri is, from the formula
given above, about 25 million million miles. Obviously
the mile is an inconveniently small unit for the
expression of so vast a distance, and in preference
astronomers use as the unit the distance which light
travels in one year, at the rate of 186,000 miles per
second, or 5*8 million million miles. Thus light takes
25 -T- 5 '8 = 4J years to come from the star to us. The
distance is conveniently expressed as 4J light-years.

The distance of our own brightest star, Sirius, is
about 8 light-years, while the pole star is about
63 light-years away. But here we are getting to
the limit of present measurements so that it is very
probable as methods improve such distances as that
of the pole star will be revised. Yet, vast as is the
distance which even light takes 60 years to traverse,
we must regard the pole star as one of our neighbours
when we compare it with other faint yet still visible
members of our system.

n] WEIGHING THE EARTH 37

CHAPTER II

WEIGHING THE EARTH1

THE Earth as a whole has no weight, if we use
weight in its strict sense of earth-pull. Corresponding
to each piece of the Earth here pulled down towards
the centre there is another piece at the antipodes
pulled up towards the centre with an equal and
opposite force, and the whole globe can be divided
into such neutralising pairs, leaving, of course, no
outstanding pull. When, therefore, we speak of
weighing the Earth we do not mean finding its
weight. We mean really finding another quantity,
the Mass of the Earth.

Let us first, then, try to make clear what the
mass of a body is and how it is related to its weight.

If we take a pound of matter, say a piece of iron
stamped as 1 lb., to different places on the surface of
the Earth, we regard it as still the same pound of
matter, wherever it is. Yet the earth-pull on it
changes its direction, and even its amount. If we
carry it from the equator towards either pole it

1 A few pages in this chapter are extracted from a paper by the
author in the Proceedings of the Birmingham Philosophical Society,
vol. ix. part i. 1893.

38 THE EARTH [OH.

gradually gets heavier, and the pull is about 1 in
200 greater near the poles than at the equator. An
ordinary balance, used in the ordinary way, will not
show this change, for it equally affects the contents of
either pan. It would undoubtedly be shown by a
spring balance if we could only get a spring at the
same time sensitive and constant in its action. But
springs are in general by no means constant or con-
sistent. They have, as it were, memories. They re-
member any change in stretch and any change in
temperature to which they have been subjected, so
that after a change and a return to the original ex-
ternal conditions their action is not quite what it was
before. It is true that their memory fades, but not
sufficiently to let us make quite consistent weighings.
There is only one kind of solid spring known which
has no appreciable memory, one made of quartz
fibre. With such a spring Mr Threlfall has succeeded
in showing change of weight with change of place.

Since, then, an ordinary balance fails, and a spring
balance is too inconsistent, to show the change in
earth-pull, how do we know that the change exists
and even what it amounts to ? Fortunately we have
an excellent detector of weight change in the pen-
dulum. If a pendulum like that of a clock is supported
so that it can swing quite freely to and fro, the time
that it takes to make one swing depends on its shape
and size and on the pull of the Earth downwards on

Ii] WEIGHING THE EARTH 39

its bob. If the pull on the same pendulum increases,
the time of swing decreases at half the rate. Now
the time of a swing can be measured with very great
accuracy, for we can watch the pendulum for hours
and count the number of swings in the total time of
watching. Dividing the total time by the number of
swings we get the time of one swing.

Pendulums have been carried about the world and
the times of swing of the same pendulum have been
exactly measured in widely different latitudes. The
results of these measurements show quite conclusively
that the weight of the bob of a given pendulum in-
creases as we travel polewards from the equator, and
we may thus describe the change. If we had a perfect
pendulum clock compensated for temperature change
and barometer change (for if the density of the air
changes, so does the effect on the buoyancy of the
pendulum change), then on removal from the equator
to this country it would gain about 130 seconds a
day, and on removal from the equator to the pole it
would gain about 216 seconds a day.

There is a change in the weight of a body not only
if we remove it north or south on the level but also if
we change its level by raising it in a vertical line.
Assuming that the pull on a body above the Earth's
surface is inversely as the square of its distance
from the Earth's centre, the weight of a body should
decrease about 1 in 2000 for a rise of 1 mile or by

40 THE EARTH [CH.

about 1 in 10 millions for a rise of 1 foot. If, then,
we have a balance as in fig. 22, p. 73, with two sets of
pans, P and Q at one level, and P' and Q' at another
lower level, and if the weights A and B exactly
balance against each other at the PQ level, they will
still exactly balance against each other if they are
both removed to the lower P'Q' level, for each gains
in weight in the same proportion. But if while B is
left in the upper pan at Q, A is taken out of P and
put into P', it alone gains in weight and the balance
will tilt down a little on the P side.

Several experiments were made in the 17th and
18th centuries to look for this change of weight
and to show it by the balance, but it was first
detected and measured by von Jolly at Munich
about 1878. He set up a balance with the pans
PQ at the top of a tower and with the lower pans
P'Q' 21 metres — say 23 yards — below. He balanced
two 5 kilogramme weights against each other at the
top. Then the weight in P was removed to P' and
the gain in weight was 32 milligrammes or about 64
in 10 millions, rather less than 69 in 10 millions given
by the inverse square law for the 69 feet change in
level. We shall describe later the device by which
the difference in air buoyancy at the two levels was
eliminated.

Some years later Richarz and Krigar-Menzel suc-
ceeded in measuring the change in weight with a

n] WEIGHING THE EARTH 41

change in level of only 2*3 metres — say 2j yards.
They used a kilogramme weight in each pan, and on
moving a kilogramme on one side from the upper to
the lower level it gained about 0*65 milligrammes or
6*5 in 10 millions, whereas the gain according to the
inverse square law should have been about 7*5 in
10 millions. In each case the increase was less than
according to the law, probably through the attraction
of the surrounding building or neighbouring elevated
ground. The law, indeed, could only be expected to
hold over the surface of the ocean.

These experiments with the pendulum and with
the balance show us conclusively that the weight of
a given piece of matter — the earth-pull on it — varies
with its situation. But there is a property or quality
which remains the same for the same matter every-
where and always. This quality is its inertia or its
mass. And the idea underlying inertia is the effort
required to get up a certain speed in the body. If
a greater effort is needed to get up the speed in one
body than in another, the first body has the greater
inertia. We give quantitative expression to the idea
by saying that an equal force is required every-
where and always to give the same rate of gain of
speed in the same piece of matter, and we say that
it always has the same mass. If there are two bodies
and we have to put double the force on to one that
we have to put on to the other for the same rate of

42 THE EARTH [OH.

gain, the first has double the mass of the second, or
generally the mass of a body is proportional to the
force needed to produce a given rate of gain of
speed1.

It is not easy 'to make exact and direct experi-
ments to test the constancy of a given piece of
matter, and the difficulty lies in applying equal
forces at different places. But the experiment is
being made for us continually, in a rather compli-
cated form, by ships' chronometers. The rate of a
chronometer is decided by the vibration of the
balance wheel against the coiling and uncoiling of
the hair spring and the weight of the wheel does not
come into account. We must suppose that, at the
same temperature, the spring offers the same resist-
ance to the same coiling wherever it may be, and as
the chronometer keeps the same time at the same
temperature in all latitudes, the rate of change in
speed of the balance wheel must be the same in a
given part of its vibration wherever it may be. In
other words the rate of change of speed under a
given force is everywhere the same, or the mass
of the wheel is constant. It may be noted that

1 It is not necessary to take into account here the interpretation
of certain recent experiments as implying a change of mass when a
body is made to change its speed. Such change could not be appreci-
able unless the speeds were enormously greater than any that we are
considering.

n] WEIGHING THE EARTH 43

we do not here suppose that we use the chronometer
at different temperatures. Our argument would fail
if we did so. For the resistance to coiling of the
spring changes with change of temperature, and
the chief aim of compensation is to correct for this
change.

A similar experiment might be made with a tuning
fork. Here again the time of vibration depends on
the resistance of the prongs to bending in or out and
only in quite negligible degree on their weight if
they are always used in the same position. Let us
suppose that a fork is tested at the same tempera-
ture in two different latitudes under conditions
which give us reason to suppose that it is the same
fork and not altered by rust or wear, and at such
small interval of time that we are entitled to take
its resistance to bending as unaltered. Then if we
find the number of vibrations per second we have
the same rate of change of motion under the same
force, or the mass is constant. Though this experi-
ment has never been made deliberately for this
purpose, it has, no doubt, often been made in the
verification of the vibration-frequency marked on
forks used in laboratories, and the constancy of mass
is verified to the same order of accuracy.

Our general conclusion from observation is that
the rate of change of motion of a body when con-
trolled by its weight changes with the place of

44 THE EARTH [OH.

observation, while the rate of change when con-
trolled by a spring, which we may fairly consider to
have constant properties, is the same everywhere.
Hence it is the weight of a body that varies and not
its mass, a conclusion which has been taken as true
for nearly 250 years and has never led to the least
inconsistency.

Newton was the first to make the idea of mass
definite, and he showed that so long as we are at
the same place the weight or the earth-pull on bodies
is exactly proportioned to their masses. If we have
one body twice the weight of another, then whatever
the bodies are, say one gold, the other wood, the first
has exactly twice the mass of the other. This could
be roughly verified by repeating Galileo's famous
experiment, in which he dropped at the same instant
various bodies over the edge of the Leaning Tower
at Pisa and showed that they reached the ground at
the same time. This meant that every pound of
weight had the same amount of mass to pull on,
whatever body the mass belonged to. In Galileo's
experiment the resistance of the air came in to
interfere with exactness. Newton used a much more
accurate test. He made two pendulums, each con-
sisting of a hollow round box, and these were hung
by strings 11 feet long so that they might vibrate
side by side. Into the boxes he put equal weights
of different substances, such as gold, silver, lead,

n] WEIGHING THE EARTH 45

glass, sand, salt, wood, wheat, and he found that the
two pendulums, if started together, continued to
swing together for a long time. The air resistance
was the same and the wood boxes were the same for
both. The only difference was the kind of matter
inside the boxes, and as the equal pulls produced
equal changes of speed for quite a long time, the
masses of the different equal weights must have
been equal. If there had been a difference, if, for
example, the gold had more mass than wood of the
same weight, the gold would have taken a longer
time for each vibration than the wood, and the
two would have got more and more out of step.
The effect being thus cumulative, would in the long
run have shown even a very small difference in
mass.

Since weight is thus shown to be exactly propor-
tional to mass, when the weighing is carried out at
the same place, we may use the balance to weigh out
different masses, and, indeed, this is precisely what
the balance does for us. The qualities of bodies for
which we purchase them are in proportion to their
mass and not to their weight. A lump of sugar has
the same sweetness here and at the equator, though it
is heavier here. A ton of coal has the same heating
power in either region, though its weight is greater
here by an amount equal to the earth-pull on six
pounds. Our ordinary description of the pieces of

46 THE EARTH [CH.

iron or brass we use on a balance as being ' weights '
does not tend to clear thought on this point. What
we term a ' pound- weight ' is really a ' pound-mass,'
and is the same wherever it may be carried about the
world. The weight of that pound varies from place
to place, and we have to remember that the weight
of a pound and a pound-weight involve different
ideas.

We can now imagine an experiment which would
give us the mass of the Earth by direct weighing —
if only it could be carried out. Let us suppose that
we could divide the whole Earth into blocks, each,
say, a cubic foot in size. Let one of the blocks be
brought up to a certain place, weighed there, and
then put back. Then let another of the blocks be
brought to the same place, weighed, and put back,
and so on until every block has been weighed. The
sum of all the 'weights' is really the sum of the
masses or is the mass of the Earth.

The experiments which we shall describe later
show that the result of such weighings would be
about 13*2 million million million million pounds or
13*2 x 1024 Ibs., a number so vast that we attach no
idea to it beyond its vastness. But the mass of
the Earth is expressed in a more thinkable way in
terms of the mass of an equal volume of water.
At the rate of 62*4 Ibs. per cubic foot this would
be about 2*4 x 1024 Ibs. Thus the average density of

n] WEIGHING THE EARTH 47

the Earth is about 13*2 x 1024^2'4 x 1024 = 5J times
that of water. Taking the density of water as 1 the
result is that the Mean Density of the Earth is about
5^, and this is the way in which its mass is always
expressed.

Though the imagined experiment would be exactly
and truly an Earth-weighing experiment, it can only
be imagined. We can make no approach to carrying
it out in practice. Our deepest mines reach down
hardly a mile, so that we make only slight scratches
on the surface, and know nothing directly of the
deeper layers.

We require, then, to measure the Earth's mass,
some other property of matter than mere earth-pull
on it and such a property was discovered by Newton
when he showed that a piece of matter is pulled not
only by the Earth but by every other piece of matter
in proportion to the mass of either piece and in-
versely as the square of their distance apart. Or,
the pull of a mass A on a mass B distant d from it,
is proportional to

Mass of A x Mass of B

which is Newton's Law of Gravitation.

Newton showed that a sphere such as the Earth,
with density the same all round at the same distance
from the centre, will pull on any outside body just as

48 THE EARTH [CH.

if all the mass of the sphere were collected into one
single point at the centre.

Now consider a body supported just above the
surface of the Earth 4000 miles from the centre.
We know that the pull on it will make it fall 16 feet
in the first second if it is allowed to drop. If we
could take it up 4000 miles, or twice as far from the
centre, and then let it drop, the law says it would

fall ~2 or - of 16 feet in the first second. If we
could take it up 12,000 miles from the centre it
would fall ^ or - of 16 feet, and so on. So that if we
could take it to 60 times the distance of the surface
from the centre or 240,000 miles it would fall ^

or ^-7. of 16 feet or just about -^ inch in the first

second.

It is just at this distance that we have a body by
which we can test the law. The moon is moving
nearly in a circle round the Earth's centre, and with
a velocity about 3400 feet per second. Let A, fig. 13,
be the position of the Moon's centre at the beginning
and B its position 3400 feet further along the curve
at the end of a particular second. Were it moving
at A free from the pull of the Earth it would move
to T, along the tangent at A, where AT is 3400 feet.

n] WEIGHING THE EARTH 49

TB is the distance it drops in the second and it
is easy to show that TB is very nearly -fa inch
or is what we expected from the law. Hence every
pound of the Moon's mass is pulled by a force
3-^ of the pull on an equal mass at the Earth's
surface.

Comparing the motions of the different planets
under the pull of the Sun, it can be shown that with
them also the pull in each case is proportioned to the
mass of the planet and to the inverse square of its
distance from the Sun. In fact a pound of mass has

Fig. 13.

on it a pull by the Sun inversely as the square of
its distance from the Sun's centre whatever the
planet of which it forms a part. So the law is
amply verified as regards the mass pulled and the
distance.

To show that the pull is also proportional to the
mass of the pulling body, we assume the law, which
holds good in all cases which we investigate, that if
two bodies A and B act on each other, the force
which A exerts on B is equal though opposite to the

P. 4

50 THE EARTH [CH.

force which B exerts on A. We shall, for simplicity,
neglect the difference of distance of different parts of
the Earth and Moon from each other. As each body
pulls the other as if it were concentrated at its
centre, this simplification is justified. The Earth pulls
the Moon with a force proportional to the mass of the
Moon, so that each pound in the Moon is pulled with
an equal force. In turn each pound pulls the Earth
with an equal force and the total is proportional to
the number of pounds of mass pulling. Thus we may
conclude that Newton's statement holds good that the
gravitation pull is proportional to the product of the
masses of the two pulling bodies.

Now we can see how the gravitative pulls of two
bodies on a third body enable us to compare their
masses. Let A and B, fig. 14, be two bodies and let
us suppose that we wish to determine the mass of A
in terms of the mass of B. Let a third body m be
distant a from A and b from B and let A pull it with
force a and B pull it with force 0.

The law of gravitation gives us

Mass of A x Mass of m
a a* =b2 Mass of A

Mass ot" B x Mass of m a2 Mass of B '

whence Mass of A — ^ • ^ • Mass of B.

II]

WEIGHING THE EARTH

51

We can for example find the mass of the Sun in
terms of the mass of the Earth. If we suppose A, B,
and m to be, respectively, the Sun, the Earth, and
the Moon, then a is 93 million miles and b is 240,000
miles. The ratio of the pulls ct/fi in the two circles,
one described in a year and the other in ^ of a year,
can be calculated and it works out to be about 1ffi.
Approximately then

Mass of Sun = 300,000 Mass of Earth.

Fig. 14.

But for the purpose of Earth weighing, A must be
the Earth, while B must be some body of which we
know the mass in pounds or kilogrammes, and we
must be able to find what is the ratio of the pulls of
the two on a third body m.

Newton discussed the possibility of comparing two
such pulls, and in two ways. In one of these he
thought of comparing the attraction of a mountain

4—2

THE EARTH

[CH.

with that of the Earth. If a plumb bob were hung
at the side of the mountain, the mountain would
draw the bob towards it. If AC (fig. 15) is the
direction in which it would hang if the mountain
were removed and if AB is its actual direction it is
easily seen that, if we consider only the horizontal
part of the mountain-pull,

Mountain-pull _ CB
Earth-pull ~ AC'

If then we can measure the angle of deflection of the

plumb line SAC we can
determine CBJAC and
therefore the ratio of the
pulls. Newton calculated
that if the mountain were
hemispherical, 3 miles
high, and of the same
density as the Earth, a
plumb bob at its base
would not be deflected
In fact, as the two attrac-
tions would be in the ratio of radius of the mountain
to the diameter of the Earth, a bob with a string a
yard long would be drawn aside -g-^v x 36 incn or TI
inch. If the actual drawing aside were \ or £ of this
we should then know that the density of the moun-
tain was \ or J that of the Earth.

Fig. 15.
so much as 2 minutes.

ii] WEIGHING THE EARTH 53

In the other way of comparing pulls Newton con-
sidered the possibility of using a sphere for B, fig. 14,
and another sphere for the mass m and he calculated
that two spheres of the density of the Earth and each
a foot in diameter, if to begin with they were £ inch
apart, would take not less than one month to draw
together into contact. There was a mistake in
arithmetic here, for the time would really only be
about 320 seconds.

Newton dismissed the subject with the remark
that in neither case would there be an effect great
enough to be perceived — a statement no doubt true
for the methods of measurement then available. But
the enormous extension of scientific theory, so largely
due to Newton, was accompanied by a great improve-
ment in the methods of measurement, and what to him
seemed impossible, was actually tried about ten years
after his death by Bouguer.

Bouguer was a member of the expedition sent
out by the French Academy to measure, as described
in the last chapter, the length of a degree of latitude
at the Equator, and, impressed by the vastness of the
Andes, he determined to try to measure the ratio of
the pull of a mountain on a plumb bob to the pull of
the Earth. For his purpose he fixed upon Chimborazo,
a mountain some 20,000 feet high, as most suitable,
and he selected a station on the south slope just
above the snow-line and about 5000 feet below the

54 ' THE EARTH [CH.

summit. Here he and his colleague, de la Condamine,
fixed their tent after a most toilsome journey of ten
hours over rocks and snow, and in face of great
difficulties due to frost and snow, they took the
zenith distances of several stars as these crossed the
meridian. Then a few days later they moved to
a second station very nearly four miles west of the
first, where the attraction of the mountain had only
a small component towards the north, not more than
T^ the value it had at the first station. Here their
difficulties were even greater than before. They
were exposed to the full force of the wind which
filled their eyes with sand and was continually on
the point of carrying away their tent. The cold
was intense, and so hindered the working of their
instruments that they had to apply fire to the
levelling screws before they could turn them. Still,
they made their observations, measuring the dis-
tances from the zenith of the same stars as they
crossed the meridian. The principle of the method
may be seen from fig. 16, where we suppose that
the stars are looked at through a telescope provided
with a plumb line hanging from its upper end.
Imagine that we begin at the second station re-
presented in the lower figure, and watch the
passage of a star which for simplicity we will
suppose to cross the meridian exactly at the zenith.
Let us suppose that at the first station the vertical

II]

WEIGHING THE EARTH

55

Ist Station

Due South of
Summit- onSlope

is deflected by the mountain. Then the same star
will appear at that station to be displaced from the
zenith towards the north. The average for different
stars was found to be about 7£ seconds. Making
corrections for the small deflection towards the north
at the second station Bouguer estimated the de-
flection of the plumb
line at the first station
to be about 8 seconds.
Had Chimborazo been
of the density of Earth,
Bouguer calculated that
it would have drawn the
vertical aside about 12
times as much, or the
Earth appeared to have
a density 12 times that
of the mountain, a re-
sult undoubtedly far too
large. But it is little
wonder that under such
adverse circumstances

Due West of
First Station

Fig. 16.

the experiment failed to give a good result. Not-
withstanding the failure, both in this experiment
and in another which we shall not describe, great
honour is due to Bouguer in that he showed that
Earth-weighing is possible. He showed that moun-
tains do really attract, and that the Earth, as a

56 THE EARTH [OH

whole, is denser than the surface strata. As he re-
marked, his experiments at any rate proved that the
Earth was not merely a hollow shell, as some had
till then held ; nor was it a globe full of water, as
others had maintained. He fully recognised that his
experiments were mere trials, and hoped that they
would be repeated in Europe.

Thirty years later his hope was fulfilled. Maske-
lyne, then the English Astronomer Royal, brought the
subject before the Royal Society in 1772, and obtained
the appointment of a committee 'to consider of a
proper hill whereon to try the experiment, and to
prepare everything necessary for carrying the design
into execution/ Cavendish, who was himself to carry
out an Earth-weighing experiment some twenty-five
years later, was probably a member of the committee,
and was certainly deeply interested in the subject, for
among his papers have been found calculations with
regard to Skiddaw, one of several English hills at
first considered. Ultimately, however, the committee
decided in favour of Schiehallion, a mountain near
Loch Rannoch, in Perthshire, 3,547 feet high. Here
the astronomical part of the experiment was carried
out in 1774, and the survey of the district in that
and the two following years. The mountain has a
short east and west ridge, and slopes down steeply
on the north and south, a shape very suitable for the
purpose.

n] WEIGHING THE EARTH 57

Maskelyne, who himself undertook the astro-
nomical work, decided to work in a way very like
that followed by Bouguer on Chimborazo, but modified
in a manner which Bouguer had suggested. Two
stations were selected, one on the south, and the
other on the north slope. A small observatory was
erected, first at the south station, and the angular
distance of some stars from the zenith, when they
were due south, was most carefully measured. The
stars selected all passed nearly overhead, so that the
angles measured were very small. The instrument
used was the zenith sector, a telescope rotating about
a horizontal east and west axis at the object glass
end, and provided with a plumb line hanging from
the axis over a graduated scale at the eye-piece end.
This showed how far the telescope was from the
vertical when it was directed to a star not overhead.

After about a month's work at this station, the
observatory was moved to the north station and again
the same stars were observed with the zenith sector.
Another month's work completed this part of the
experiment. Fig. 17 will show how the observations
gave the attraction due to the hill. Let us for the
moment leave out of account the curvature of the
Earth, and suppose it flat. Further, let us suppose
that a star is being observed which would be directly
overhead if no mountain existed. Then evidently at
S. the plumb line is pulled to the north, and the

58

THE EARTH

[CH.

zenith is shifted to the south. The star therefore
appears slightly to the north. At K there is an
opposite effect, for the mountain pulls the plumb
line southwards, and shifts the zenith to the north ;
and now the same star appears slightly to the south.
The total shifting of the star is double the deflection
of the plumb line at either station due to the pull of
the mountain.

z.f

Soufh
Shhon

Fig. 17.

But the curvature of the Earth also deflects the
verticals at N". and S., and in the same way, so that
the observed shift of the star is partly due to the
mountain, and partly due to the curvature of the
Earth. A careful measure was made of the distance
between the two stations, and this gave the curvature
deflection as about 43". The observed deflection was

ii] WEIGHING THE EARTH 59

about 55", so that the effect of the mountain, the
difference between these, was about 12".

The next thing was to find the form of the
mountain. This was before the days of the Ordnance
Survey, so that a careful survey of the district was
needed. When this was complete, contour maps
were made, and these gave the volume and distance
of every part of the mountain from each station.
Hutton was associated with Maskelyne in this part
of the work, and he carried out all the calculations
based upon it, being much assisted by valuable sug-
gestions from Cavendish.

Now had the mountain had the same density as
the Earth, it was calculated from its shape and dis-
tance that it should have deflected the plumb lines
towards each other through a total angle of 20*9", or
14 times the observed amount. The Earth, then, is
If times as dense as the mountain. From pieces of
the rock of which the mountain is composed, its
density was estimated as 2 J times that of water. The
Earth should have, therefore, density If x 2J or 4£.
An estimate of the density of the mountain, based on
a survey made thirty years later, brought the result
up to 5. All subsequent work has shown that this
number is not very far from the truth.

An exactly similar experiment was made eighty
years later, on the completion of the Ordnance Survey
of the kingdom. Certain anomalies in the direction

60 THE EARTH [CH.

of the vertical at Edinburgh led Colonel James, the
director, to repeat the Schiehallion experiment,
using Arthur's Seat as the deflecting mountain. The
value obtained for the mean density of the Earth was
about 5J.

Experiments have also been made in which the
attraction of a part of the Earth's crust such as a
mountain, or the layers above the bottom of a mine,
has been compared with that of the whole Earth
by its effect in altering the time of vibration of a
pendulum. This method was employed in Bouguer's
second experiment mentioned above. But it has
never yielded satisfactory results. Indeed it is now
recognised that, in common with the method of the
deflection of the vertical by a mountain, it is not very
trustworthy. For in the first place there is inevit-
able uncertainty in the density of the part of the
crust used. Even if we knew the density of Schiehal-
lion exactly, there is ignorance of the density of the
strata underneath. Often there appears to be a
defect in the attraction which might be expected to
arise from tablelands and mountain ranges, and Airy
made a suggestion that these raised masses may be
buoyed up, like the peaks of icebergs, by lighter
matter below. In cases of very ancient and sinking
rocks there may be heavier matter below. In the
second place, in calculating the effect of a mountain
we must take into account the attraction of other

ii] WEIGHING THE EARTH 61

raised matter in the neighbourhood and it is a ques-
tion how far we are to go. Hutton in the Schiehallion
experiment stopped at 3 miles. But a mass eight
times as great at 6 miles would have an equal dis-
turbing effect and any large raised mass at the greater
distance should be taken into account. For these
reasons probably, the results of the various experi-
ments in which a ' Natural Mass' has been used,
such as a mountain or the Earth's upper strata, have
varied over a considerable range.

We turn now to the second method of experiment
considered by Newton, in which is measured the
attraction between two spheres, each of known size
and at a known distance from centre to centre. This
we may call the ' Prepared Mass ' method. Let us
suppose that we find that a sphere of mass M attracts
another sphere of mass m with a force P when their
centres are d apart and that the Earth of mass E and
radius R attracts m with force W, its weight. As-
suming that the Earth attracts as if it were all
collected at its centre we have

E_.M_

W d?-

Then »*

We have then to find the pull P at distance d due to
the mass M on a sphere of weight W.

62 THE EARTH [OH.

The idea of making such an experiment occurred
towards the end of the 18th century to the Rev. John
Michell, the discoverer of the inverse square law of
magnetic action. The 'torsion balance' for the
measurement of forces such as those between mag-
netic poles was invented independently by Michell
and by Coulomb. It consists of a horizontal rod
suspended from its centre by a thin wire or fibre
which resists a twist. The force to be measured is
then applied at one end of the rod in a horizontal
direction and at right angles to the rod, and the rod
is pulled or pushed round by the force. If the force
is very small, the angle of twist is usually small, and
its measurement enables us to find the force. Michell
saw the possibility of measuring the gravitative pull
between masses not too large to handle or move, and
constructed some apparatus for the purpose. He died
in 1793 without making any experiments with it and
after his death the apparatus came into the hands of
Cavendish, the great chemist and physicist, one of
whose achievements was the discovery of the consti-
tution of water.

Cavendish reconstructed most of the apparatus,
and in the years 1797-8 he carried out the great
Earth- weighing experiment known as the Cavendish
experiment. Though the idea was due to Michell it
is right that Cavendish's name should be attached to
the work, for the details both of the apparatus and of

II]

WEIGHING THE EARTH

63

the mode of using it are due to him, and he made the
experiment in a manner so admirable that it marks
the beginning of a new era in the measurement of
small forces.

Cavendish sought to measure the pull between a
lead sphere 12 inches in diameter weighing about

Fig. 18. Cavendish's Apparatus. Elevation.

hh, torsion rod. xx, balls hung from its ends. WW, attracting masses
movable round axis P. TT, telescopes to view position of torsion rod.

350 Ibs., on a lead sphere 2 inches in diameter and
weighing about 1 Ib. 10 ozs. when the distance between
their centres was about 9 inches. The apparatus is
represented in elevation in fig. 18 and in plan in fig. 19.
It was enclosed in a chamber GGGG built within

64 THE EARTH [OH.

another to ward off changes of temperature, and the
air currents thereby produced. The torsion rod hh
was of deal 6 feet long tied by wires hg to an upright
mg to give strength and rigidity. In order to double
the effect there were two attracted 2 inch spheres xx
hung by short wires from the ends of the rod and the
rod itself was hung by a wire Ig of silvered copper
about 40 inches long from the top of the protecting
case at F. There were also two attracting spheres
WW, each 12 inches in diameter, hung from a cross
piece as shown in fig. 18 and these could be moved

Fig. 19. Cavendish's Apparatus. Plan.
Attracted balls x^x^. Attracting masses W^W^.

from the positions JFiTF2 in fig. 19 to the positions
w-flD* round an axis coinciding with the axis of the
wire Ig, by a cord passing outside the enclosing
chamber at m.

The position of the torsion rod was determined by
a mark (really a vernier) on the end of the rod which
moved over a divided scale fixed near the end. The
scale was lighted by a lamp L and viewed by a
telescope T.

n] WEIGHING THE EARTH 65

In fig. 18 the two attracting spheres WW are not
in position for exercising the maximum pulls on xx.
They would have to be moved round a little further
to give the positions TPiTFg of fig. 19. They were
stopped in the latter position by pieces of wood when
J inch from the case and with just under 9 inches
from centre of W to centre of x.

The apparatus was to give P, the pull of W on x
at 9 inches. Imagine that we begin with the spheres
WW far away (or, what is equivalent, in the line at
right angles to the torsion rod through its centre)
and that we read the position of the end of the rod
on the scale. Now bring the masses into the posi-
tions TFiTT2, fig. 19, when there is a pull P at each
end of the rod, turning the rod round, and we observe
that it turns through n divisions of the scale. Then
move the masses round to the positions wtfjo* when
the two pulls P are reversed and the rod moves
round through n divisions from its first position in
the opposite direction. The total change of reading
of the rod on the scale between the WiW2 and the
u\w2 positions of the masses will be 2n and it will be
unnecessary to observe the reading when the masses
are half-way between, the equivalent to being far
away. The deflection of 2n divisions is equal to the
deflection which would be produced by 4P applied
at one end only.

The next thing is to find the actual force

p. 5

66 THE EARTH [CH.

corresponding to the observed number of divisions on
the scale. When a system of this kind is suspended
by a wire so that it can vibrate in a horizontal plane,
twisting and untwisting the wire, the time of one
vibration to and fro is the same whatever the extent
of the excursion and depends on the arrangement of
the mass of the system about the axis of vibration
and on the force, applied at the end of the arm,
needed to twist the wire through unit angle. If then
we observe the time of vibration and know how the
mass is disposed, we can find the force which will
twist the system through the unit angle. But the
force for any other angle is in proportion to the
angle, so that we can calculate the force 4P needed
to twist through 2n divisions, or the force P twisting
through \n divisions.

We may put this into simple mathematical form
if we neglect all corrections. Suppose that the
2n divisions correspond to an angle of deflection 6,
and that the torque per radian twist of the end of
the wire is /*. Then if a is the arm at which P acts,

4Pa = //,0 (1).

If / is the moment of inertia of the vibrating
system round the axis of the wire and if T is the time
of one vibration,

n] WEIGHING THE EARTH 67

P ^16
whence P — ^™~ >

and going back to the formula on p. 61, in which M
now represents the mass of one of the attracting
spheres WW (fig. 18), W is the weight of one of the
attracted spheres xx, and d is the distance of the
centres apart,

^__ ,.

~ **

As we now know the quantities on the right hand
of (3), we have determined E the mass of the Earth.

We have supposed in this account that only the
masses W acted on the masses x, but in reality the
rods suspending the masses exercised some attraction,
and both masses and rods exercised some attraction
on the torsion rod M. Further, each mass was
attracting not only the ball nearest to it but also to a
small extent the further ball, and all these attractions
had to be taken into account and allowed for, the
observed value of P being the value used with
formula (1) multiplied by a certain factor which could
be determined from the arrangement and dimensions
of the apparatus.

Cavendish made 29 separate determinations, and
the value for the mean density of the Earth resulting
from these determinations is 5*448. This is corrected
for a mistake which was detected in the original
paper many years after its publication.

5—2

68 THE EARTH [CH.

The experiment has been repeated several times
since by other workers and the most notable repetition
is that by Professor C. V. Boys, who published an
account of his experiment in 1895.

Boys had a few years before invented a method of
drawing out fibres of quartz of great fineness — a
diameter of y^^ inch being quite easily obtained.
He found that these fibres are extraordinarily strong
for their diameter and extraordinarily true in their
elastic properties. A quartz fibre may be twisted
round very many turns and, on being released, it will
untwist the same number of turns and come back, as
nearly as can be determined, to its original position ;
whereas a metal wire thus twisted acquires 'per-
manent set/ and on release does not untwist the
whole way back to the original position. By this
great invention Boys put into the hands of physicists
a means of making torsion balances for the measure-
ment of small forces far exceeding in delicacy and
accuracy anything hitherto used. He determined to
repeat the Cavendish experiment, using a quartz fibre
instead of a metal wire to suspend the torsion rod.
It was necessary to reduce the size of the vibrating
system to be small enough to be carried by a fine
fibre. This reduced the sizes and distances to be
measured and it was perhaps more difficult to measure
these sizes and distances with proportionate accuracy ;
but, on the other hand, the twist for a small force
increased with the finer fibre and the apparatus

n] WEIGHING THE EARTH 69

became so small that it could be kept at a much more
uniform temperature, and air currents, which are in
a closed case entirely due to uneven temperature at
different parts of the case, were thereby very greatly
reduced. These air currents are the chief disturbers
in such an experiment and were found to be very
troublesome in the larger apparatus used by
Cavendish.

The suspending fibre which Boys used was about
17 inches long, and probably about j^ inch in dia-
meter. The torsion rod was only ^ inch long in place
of Cavendish's 6 foot rod. The attracted spheres at
its ends were in one set of experiments gold balls
J inch in diameter, and the attracting spheres were
lead 4J inches in diameter. The torsion rod was
itself a mirror and the image of a divided scale
22 feet away was viewed in the mirror by a telescope.

If the attracting and attracted masses had all
been on one level as in Cavendish's experiment, it
will be seen from the plan in fig. 21 that with a
distance of less than an inch between the attracted
masses a 4^ inch sphere in front of one mass would
have been almost equally in front of the other and
with nearly the same distance between centres, and
so, pulling them almost equally in the same direction,
would not have tended to turn the rod round much.
Boys therefore adopted the plan represented in eleva-
tion in fig. 20. The attracted masses were suspended

70

THE EARTH

[CH.

by quartz fibres from the ends of the mirror torsion
rod at different levels, one 6 inches below the other,
and one of the attracting balls was on each of these
levels.

Fig. 20.

The attracted balls hung in an inner protecting
tube and the attracting balls hung in an outer case
which surrounded the inner tube and could be re-
volved round it. Fig. 21 represents a plan on which
the centres mz and M 2 must be supposed to be 6 inches

II]

WEIGHING THE EARTH

71

below the plane containing the centres of mi and
MI. As the case containing the attracting balls was
revolved there was a position M^M^ in which the
moment of the pulls on the attracted balls m^m* was
a maximum in one direction and a position MiM2'
in which it was a maximum in the other direction.
Thus each attracting mass acted in both its positions
on the same attracted mass. The general theory of
the experiment is like
that of the Cavendish
experiment and we need
not repeat it. The final
result of Professor
Boys's work gave the
mean density of the
Earth as 5*527, and for
the present this may be
taken as the most trust-
worthy result.

We have now to de-
scribe another mode of
experiment, in which the pull between two masses is
measured by the common balance instead of by the
torsion balance. Though the common balance is in
some ways less satisfactory for the purpose, it is
well in work of this kind, where the quantity to be
measured is small, to have different modes of attack.
For there might possibly be some undetected error,

Fig. 21.

72 THE EARTH [CH.

characteristic of one method, which a divergence of
result by another method would reveal. An agree-
ment by the two methods gives us confidence in both.

The first account of a common balance experiment
was published by the late Professor von Jolly of
Munich in 1878. In his final work, a little later,
a balance was mounted on a support at the top of
a tower, with scale pans under the two ends of the
beam in the usual position. Another pair of scale
pans was suspended by wires from these, 21 metres,
say 23 yards, below, nearly at the bottom of the
tower as represented in fig. 22. Four glass globes
A BCD of equal weights and volumes were prepared
and two of them, A and B, were filled each with
5 kgm. of mercury. Then all four were sealed.
First A and B were put in the upper pair of pans,
and C and D in the lower pair, and a balance was
made. Then A and C were interchanged. The
equality of volume of the two globes eliminated any
effect due to the greater buoyancy of the air below
and there was a gain in weight rather more than
31 mgm., due to the approach of the 5 kgm. of
mercury to the Earth. This is the first experiment
in which a change in the weight of a body in so
small a change in height as 21 metres was demon-
strated.

A lead sphere about 1 metre in diameter was now
built up out of separate blocks, immediately under

n]

WEIGHING THE EARTH

73

one of the lower pans. On again effecting the inter-
change between A and C, A when brought below
weighed 0*59 mgm. more than it did before, and this
was the pull on it by the lead sphere. The distance

21m

Q

Floor at bottom.
Fig. 22.

from centre of lead to centre of mercury was about
57 cm. If, then, the lead sphere at an effective
distance of 57 cm. exercised a pull of 0'59 mgm. on

74 THE EARTH [CH.

the mercury, and the Earth at an effective distance
equal to its radius, about 690 million centimetres,
exercised a pull of 5 kgm. or 5 million milligrammes,
the mass of the Earth could at once be calculated
in terms of the mass of the sphere of lead. When
the result was put in the usual way, the mean density
of the Earth came out as 5 '69.

An experiment on similar lines was carried out
later by Richarz and Krigar-Menzel. Like von Jolly,
they had a balance Avith pans at two levels, but their
change in level was only 2*3 metres. They used two
solid spheres each weighing 1 kgm. and two hollow
spheres of the same external volume as these and
weighing 53 gm. each. Virtually they began with a
solid sphere above and a hollow sphere below, on the
left say, and the reverse arrangement on the right.
Then on each side solid and hollow were inter-
changed, and the left gained while the right lost by
the interchange. The effect observed was there-
fore twice the effect of the change in level of
1000 - 53 = 947 grammes. They found that the effect
of lowering 1 kgm. 2*3 metres was a gain in weight
of 0'65 mgm.

A rectangular block of lead about 2 metres high
and nearly cubical was then built up of separate
pieces under the balance and between the two levels.
There were narrow vertical tunnels through the
block for the passage of the wires to which the lower

n] WEIGHING THE EARTH 75

pans were attached. When, starting with solid above
and hollow below on the left and with the reverse
on the right, an interchange was made of solid and
hollow, the left-hand solid had the attraction of the
lead changed from a pull down to a pull up, while
the right hand had the reverse change. The effect
of the interchange was therefore that of change in
height minus four times the pull of the lead block
on one sphere. The experiment gave the attraction
of the lead block on one sphere as 0*36 mgm., whence
the mass of the Earth could be found in terms of
the mass of the lead. The mean density of the Earth
deduced was 5*505.

About the same time that von Jolly began his
experiment the author also saw the possibility of
using the common balance to measure the attraction
between two masses and made some preliminary
trials which ultimately led to an experiment, which
was carried out at Birmingham. As the author
knows more about this experiment than about the
other experiments by the common balance, it is
selected for more detailed description.

The balance (fig. 23) was of the type used at mints
to weigh out bullion. It had a specially strong beam
4 feet long. It was supported on two iron girders,
seen in section in gg, and these were supported on
two brick pillars, of which the one at the back only
is shown. In order to prevent the vibrations due

76

THE EARTH

[CH.

to street traffic and to the shutting of doors in the
building one course of brickwork in each pillar was
replaced by a number of indiarubber blocks. The

Fig. 23.

A A, weights, each about 50 Ibs., hanging from the two arms of
balance. M, attracting mass on turn-table, movable so as to
come under either A or B. TO, balancing mass. A'B', second
positions for A and B. In these positions the attraction of M on
the beam and suspending wires is the same as before, so that
the difference of attraction on A and B in the two positions is
due to the difference in distance of A and B only, and thus the
attraction on the beam, &c., is eliminated.

balance was enclosed in a large wooden case, lined
inside and out with tinfoil, the metal surface re-
flecting radiation falling on it from outside and

n] WEIGHING THE EARTH 77

radiating little to the inside, and so lengthening
out and reducing fluctuations of temperature. The
apparatus was in a closed cellar and the tilt of the
balance beam was observed by a telescope through
a hole in the floor of the room above.

The pans of the balance were removed and in
their place two lead spheres A and By each 6 inches
in diameter and weighing about 21 '6 kgm. or 48 Ibs.,
were hung from the ends of the beam. These were
the attracted masses. The beam was not lifted up
from its support between weighings, as in the usual
operations with a balance, but was left free to swing
through a whole series of experiments, often ex-
tending over a number of days.

Underneath the balance was the attracting mass
M, I foot in diameter and weighing 153*4 kgm. or
340 Ibs. This was placed on a turn-table which could
be rotated about an axis exactly under the centre
of the balance by a rope passing to the observer
in the room above. M could be brought against
a stop so as to be exactly under A, with a distance
of 1 foot from centre to centre, or it could be
moved round against another stop so as to be
exactly under B.

The attraction of M on A in the first position
made A slightly heavier. When it was moved round
to the second position under B, its attraction was
taken from A and added to the weight of J5, and

78

THE EARTH

[CH.

Fixed Bracket Bracket.

the balance tilted over on the B side through an
exceedingly small angle due to a change in the
weight of B amounting to twice the attraction to be
measured. It was necessary, then, to measure the

small tilt and to find the
change in weight, or the
attractive pull, to which
it corresponded.

Firstly, to measure
the tilt a ' double-sus-
pension ' mirror was used,
a device due to Lord
Kelvin. This was applied
as shown in fig. 24. The
beam of the balance must
be supposed to be per-
pendicular to the plane
of the figure some 2 feet
above the end of the
Vanes working pointer. Near the end

in dashpot.

Fig. 24.

tion was a fixed bracket.

of the pointer a bracket
was attached to it, and
opposite to its mean posi-
A mirror was hung from
the ends of these brackets by two silk threads. Now
imagine that the balance beam tilts, say the further
end downwards. The pointer will move out of the
plane of the paper towards us and the mirror will

n] WEIGHING THE EARTH 79

turn round, and the angle through which it turns will
be as many times greater than the angle through which
the beam turns as the length of the pointer is greater
than the distance between the suspending threads.
The length of the pointer was about 150 times this
distance, and the advantage of such magnification is
obvious, as the tilt of the beam was not much more
than a second of arc. To prevent the swinging of the
mirror independently of the balance a set of vanes
was attached to it below, working in a dashpot con-
taining mineral oil. An inclined mirror, not shown
in the figure, was fixed just in front of the suspended
mirror. An illuminated scale was fixed to the tele-
scope, and the light from this was reflected first from
the inclined mirror to the suspended one, then back
to the inclined one, and so up into the telescope.
The observer saw the image of the scale moving up
and down as the balance moved and noted the
division on a cross-hair which was fixed in the eye-
piece in the middle of the field of view.

Secondly, to determine the weight-value of the
tilt two riders were used, each 1 centigramme in
weight. One of these was lifted off the beam and
the other was put on to it exactly 1 inch further
from the centre, equivalent to a transfer of the first
rider through 1 inch. As the half length of the
beam was 24 inches, this was equivalent to an
addition of -fa of 10 mgm., about 0*42 mgm., at the

80 THE EARTH [CH.

end of the beam. The change in the scale-reading
due to the change of rider was noted. It happened
that it was very nearly equal to the effect of moving
M from its position under A to its position under
Bj so that the pull of M on A at 1 foot was about
0*21 mgm. It is not necessary to describe the
method of putting the riders on and off, but it
may be mentioned that in order to secure a transfer
of exactly 1 inch two little frames equivalent to
two little scale pans hung from points equivalent to
knife edges exactly 1 inch apart along the beam.

We shall not give particulars of the weighings.
It will suffice to say that observation of the change
in scale-reading due to shift of the rider was alter-
nated with that due to change of position of M for
a considerable number of determinations of each,
and the means were taken. The mass M not only
attracted the hanging masses A and B, but also
their suspending rods and the arms of the balance.
To get rid of these effects, a second set of measure-
ments was made, in which A and B were put higher
up the suspending rods in the positions A and B',
so that there was double the distance, viz. 2 feet,
from centre to centre. The attractions on A and
B were reduced to \ the previous amount, but the
attractions on beam and rods remained as before.
The difference between the two values was thus f the
value of the attraction in the lower position.

n] WEIGHING THE EARTH 81

Of course there were cross attractions of M on B
when it was under A and of M on A when it was
under B, tending to reduce the effect observed.
But the reduction could be calculated and allowed
for.

Originally the mass M alone was on the turn-
table, but some curious inconsistencies appeared in
a series of results obtained, and ultimately it was
found that these inconsistencies were due to a tilting
of the cellar floor when the mass M was moved from
one side to the other. The floor probably tilted
through an angle about a third of a second, which
would amount to 1 inch in 10 miles, and this tilt
was quite enough to affect the results very seriously,
as the whole tilt of the beam due to the change in
attraction when M was moved round did not amount
to as much as 2 seconds. The floor-tilt had been
looked for before exact measurements were begun,
but it had not been detected. It asserted its exist-
ence later, and in such a way as to spoil a long series
of measurements. It may be noted that if the
tilt had always been the same, it would have been
eliminated by the differential method of taking the
attraction. But it grew as time went on, for the
floor gradually settled down and became more com-
pact, all tilting over together.

In order to prevent any tilt a second mass ra
(fig. 23) was introduced, having half the weight of M

p. 6

82 THE EARTH [OH.

and being at double the distance from the centre
on the opposite side. The centre of gravity of the
two was thus at the centre and the prevention of
tilt of floor was complete. The attraction of m on
A and B had now to be allowed for, but that only
made the calculation of the results a little more
complicated.

A very rough value of the mass of the Earth may
be obtained thus : M attracted A, at an effective
distance of 1 foot, with a force of 0*21 mgm. weight.
The Earth attracted A at an effective distance equal
to its radius of 21 million feet, with a force equal
to the weight of A, i.e. equal to 21 kgm. or 108 times
as much. Had the Earth been 1 foot away its mass
would have been 108 x mass of M or 108 x 340 Ibs.
But as it was 21 million feet away its mass was
(21 x 106)2 times this or about 15 x 1024 Ibs. — an over
estimate due to inexact numbers and neglect of
corrections. As the mass of an equal sphere of
water is about 2'5 x 1024 Ibs. the mean density of the
Earth is roughly 6.

The result obtained for the mean density after all
corrections was 5 '49.

It may be interesting to state the accuracy with
which the balance worked. The increase in the
weight of the 50 Ibs. which was to be measured
was about eo.ooVooo of the wnole weight. Measure-
ments of this increase were never wrong by more

ii] WEIGHING THE EARTH 83

than 2 per cent, of the amount, usually well within

1 per cent., or 6,00o.ooo.ooo of the whole weight, the
variation which would occur if the 50 Ibs. were moved
•fa inch nearer to the centre of the Earth. Now these
numbers in the denominator are too large to give us
much idea of the smallness of the weights concerned.
Suppose, then, we take a rough illustration, in which
the small weights are magnified up to be appreciable.

Imagine a balance large enough to contain on
one pan the whole population of the British Islands,
and that all the population has been placed there but
one medium-sized boy. Then the increase in weight
which had to be measured was equivalent to mea-
suring the increase due to putting that boy on with
the rest. The accuracy of measurement was equi-
valent to observing from the increase in weight
whether or no he had taken off one of his boots
before stepping on to the pan.

One of the most curious points about this method
of weighing the Earth is the contrast between the
mass to be weighed and the mass in terms of which
it is weighed. It will be remembered that the tilt of
the balance was measured by moving a centigramme
rider along the beam. Any inaccuracy in the esti-
mation of the weight of that rider is repeated in the
weight of the Earth. So that in one sense we may
be said to weigh the Earth with its 13 billion billion
pounds by using a weight of 50|1000 part of a pound.

6—2

84 THE EARTH [CH.

The results of all recent experiments, whether by
the torsion balance or by the common balance, agree
in giving to the mean density of the Earth a value
very near to 5*5, and probably the real value is a
little greater than this, but not so much as 5 '55.

Though all the experiments have been described
as if they were designed to find the mean density of
the Earth, they have a more general aspect and may
be regarded as determining the exact expression of
Newton's Law of Gravitation. That law states that
the attraction between mass Ml and mass M2 a

7 , • A. i , Ml x MS T

distance d apart is proportional to - ^ — £. Let

(MJ

the masses be measured in grammes, the distance
in centimetres and the attraction P of either on the
other in dynes. We may put the law in exact form
as

GxM.x M,
d2

where G is a constant, the same for all masses,
whether they be the Sun and Earth or the Earth
and Schiehallion or the attracting and attracted
spheres in any of the Cavendish class of experiments.
It is called the Gravitation Constant.

Now as any of the Cavendish experiments con-
sists in determining P between known masses Ml
and Jf2, d apart, the result gives us G at once. It

in] THE EARTH AS A CLOCK 85

may be shown that the other Earth-weighing ex-
periments also give 6r, though not quite so directly.

The value of G is very near to

CHAPTER III

THE EARTH AS A CLOCK

EVERY day the telegraph lines over the whole
country cease work for a short time for the passage
of a signal which is sent out from the Observatory
at Greenwich exactly at 9 a.m. At the Observatory
there is a Standard Clock, and that Standard Clock
is the Earth itself. The sky is the dial. Its figures
are the stars, and the line of sight of a telescope is
the hand which points the hours. What sort of time
does this clock keep ?

If we face southwards on a clear day we note
that the Sun has risen on our left, mounts to the
highest point in the south, and sinks down to set on
our right. On a clear night we note that if we still
face southwards the Moon and stars move in the
same way from left to right. We feel at rest, and we
see the lights of day and night moving over us.

86 THE EARTH [CH.

Till 300 or 400 years ago almost everyone
believed that this was the only and final account
of the appearance, that the Earth was at rest and
that the sky moved round. But now we are certain
that a more convenient and therefore a better
account is that the appearance is due to the Earth
turning round under the sky. It is no wonder that
this new account had a hard fight against the old
belief, and that it only slowly conquered. It is
impossible to realise that we are being whirled
round in a huge circle, travelling in this latitude
of Britain at a speed of 600 miles an hour. It seems
at first thought as if we should be whirled off into
space. But it is easy to show that a very minute
fraction of our weight is sufficient to keep us from
so flying off".

The feeling that we are at rest while the sky is
moving over us is just like that which we have when
we are seated in a smoothly running train and see
the buildings and telegraph posts rushing past us ;
or when we are in the cabin of a steamer on a river
and, looking out of a porthole, see the river banks
drifting past us. If our only aim is to describe what
we 'see in change of relative position, it is perfectly
correct to say that we are sitting still in the train
or boat and that the country is moving past us.
And in a similar way it is perfectly correct to say,
if we are only describing the relative change of

in] THE EARTH AS A CLOCK 87

position, that the Sun, Moon and Stars rise in the
east, climb up the sky and set in the west.

But when we come to consider not only change
of relative position but the forces which effect the
change, then we are obliged to think of one de-
scription as better than the other. Our train stops
at a station, and we can think of the friction of the
rails against the braked wheels as stopping it. We
cannot think of a force which would stop the station
at the train. Our steamer stops at a pier, and we
can think of the pulls of the mooring ropes acting
to stop it. It is impossible for us to think that the
whole countryside is pulled up alongside the steamer.
Similarly with the Sun, Moon and stars. We know
that the force needed to keep a body moving in a
circle is proportional to its distance and to the
square of its rate of revolving round the centre.
We can easily think of the weight of bodies as
sufficing to keep them on the surface of the Earth
if it whirls round once in 24 hours. But if the Earth
were at rest and the heavenly bodies were moving
round it once in 24 hours, it would indeed be diffi-
cult to imagine forces big enough. Each pound of
matter in the Moon, to get round a circle 240,000
miles in radius in 24 hours, would require a pull
of about J Ib. weight. Each pound in the Sun,
to get round in its circle, would require a pull of
about 80 Ibs. weight ; each pound in Sirius, to get

88 THE EARTH [OH.

round in its gigantic circle, from which light only
comes to us in 8J years, would require a pull of
something like 20,000 tons. We cannot contemplate
the possibility of these huge forces : so that unless the
Sun, Moon and% stars are mere phantasms and not
real matter, we are obliged to think that the motion
is in the Earth and that it is whirling round under
the sky.

We have direct evidence of this whirling. The
shape of the Earth as described in the first chapter
is a consequence. Suppose that a perfect sphere the
size of the Earth were suddenly started spinning
once in 24 hours round an axis. Part of the weight
of the matter in the equatorial regions would be
used up, as it were, in keeping it moving in its circle.
It would press less towards the centre. The matter
at the poles, not moving round, would press with its
whole weight, with the result that it would press
out the equatorial matter and make it bulge, and
an equatorial bulge is just what the shape of the
Earth shows.

Another consequence is the direction of spin in
cyclones, the vast whirlwinds which are such
common features of the weather in this part of the
globe. The centre of a cyclone is a point where
the barometric pressure is lower than anywhere
in the neighbourhood and the wind circles round
the centre always in the same direction, counter-

in] THE EARTH AS A CLOCK 89

clockwise as plotted on a map in the northern
hemisphere, clockwise in the southern hemisphere.

Let C (fig. 25) be the centre of such a cyclone,
the point of lowest pressure. Let the circle NWSE
be a line on the map passing through points round C
at which the pressure has a certain higher value.
At first thought we might expect the air to be
pressed straight in towards C from all sides. But
the wind does not blow straight in to the centre

N

•c

f

8

Fig. 25. Fig. 26.

of low pressure. It is observed much more nearly
to circle round it. In general it is inclined some-
what inwards, as indicated by the arrows in fig. 25,
and always in our latitude the whirling is counter-
clockwise.

The way in which cyclones are formed is not yet
understood, but the following explanation of the
direction of the whirling is probably correct. Con-
sider a mass of air at 8 (fig. 26) to the south,
which tends to move to C as it moves northwards.

90 THE EARTH [OH.

It keeps moving into regions travelling less rapidly
to the east than the regions from which it has
come. It keeps some of the excess of its W. to
E. motion, and so instead of moving due north to C
it moves partly to the E. or, on the whole, to the
N.E., and so may have the direction of the lower
arrow, fig. 26.

Next consider the motion of a mass of air from
N. As it moves southwards it will be continually
moving into regions with a greater W. to E. motion
than its own. It will lag behind and come partly
from the E. as well as the K, and so may have the
direction of the upper arrow (fig. 26).

Thus there is imparted a tendency to whirl round
the centre and always in the same direction ; the
winds tending to go to the right of the centre, and so
starting a counter-clockwise rotation to the cyclone.

If the case of the Southern hemisphere be con-
sidered it will easily be seen how it comes about that
the whirling of the cyclones there is in the opposite
or clockwise direction.

The shape of the Earth and the whirling of
cyclones constitute observational evidence for the
rotation of the Earth. We owe to Foucault1 two
experimental methods of proving the rotation. The
first of these is that of the Foucault pendulum.
To understand it let us assume that the Earth is

1 Recueil des Travaux Scientifiques de Leon Foucault.

Ill]

THE EARTH AS A CLOCK

91

spinning round the polar axis POP (fig. 27). Take
any point A on the surface and draw the two lines
OA and OB through the centre 0 at right angles
to each other, and in a plane through POP ; we may
resolve the spin about OP into two spins going on
at the same time about OA and OB.

It is important to note that the spin of a body
round an axis can be represented by a length along
that axis proportional to the
number of turns in a given
time. The rule for resolving
a spin is as follows. If OP is
taken to represent the rate of
spin round PP, once in 24
hours, dropping perpendicu-
lars PM on OA and PN on
OB, the spin round OA is re-
presented by OM, while that
round OB is represented by
ON. In general OM and ON are less than OP,
so that the spins which they represent are slower
than once in 24 hours. Thus the surface of the
Earth at A is being carried forward, out of the
plane of the figure, by the motion round the axis OB,
while at the same time it is turning round the
vertical OA with a spin OM. When A is at the
pole this turning around the vertical is a maximum,
for there OM = OP, and a revolution is effected in

Fig. 27.

92 THE EARTH [OH.

24 hours. As we move A down towards the equator,
OM decreases. In our latitude the ground turns
round the vertical once in about 31 hours. When
A is at the equator, OM vanishes and there is no
turning round the vertical.

Foucault's experiment consists in hanging up
a heavy pendulum by a wire many feet long and
setting it swinging in a definite vertical plane. As
the surface of the ground and, with it, the support
of the pendulum turn round the vertical, the only
action on the pendulum is slowly to twist the wire,
and this merely twists the bob round its vertical
axis. It goes on swinging in the plane in which it
was started, and the ground revolves underneath it
in a counter-clockwise direction.

But to the observer moving round with the
ground the plane in which the pendulum bob swings
appears to move round clockwise as looked at from
above. With a pendulum some 30 feet long, swing-
ing across a horizontal circle of 4 feet radius drawn
round its lowest point as centre, the bob moves, each
time of return, to a point about ^ inch further
round the circle so that the motion is evident in
quite a few swings.

Foucault's second mode of showing the rotation
of the Earth was by means of the gyroscope, a heavy
disc which can be set spinning on specially arranged
supports.

in] THE EARTH AS A CLOCK 93

When a body is acted on by a force through its
centre of gravity the force does not tend to turn the
body round, but merely to move it forward. If the
force is in a line passing to one side of the centre of
gravity, a 'side way' force as we may term it, then
it tends to spin the body round as well as to move it
on. But when the body is already spinning round
an axis the effect of the sideway force in changing
the direction of that axis is less the greater the spin.
For instance, let a force act to one side of the centre
of gravity of a sphere, which in one second will give

Fig. 28.

a spin round OA, fig. 28, represented by OA. If the
body was previously at rest, a line OX will revolve
round OA at a rate proportional to OA. But now
let the body be revolving round OX with a spin re-
presented by OX. Then we must compound the two
spins OA and OX, and their resultant is OB, or the
new spin is about OS, and at a rate represented by
OB. The greater 0 X is in comparison with OA the
nearer OB is in magnitude and direction to OX.

The gyroscope which Foucault used was arranged
as represented diagramatically in fig. 29. D was a

94

THE EARTH

[CH.

heavy disc whose axis was pivoted on a circle (7(7.
This circle was supported in turn by knife edges KK,
on which it was exactly balanced, and these rested
in V-shaped hollows on a second circle EE. This

circle was suspended by a silk fibre SS. The disc,
then, could be turned into any position subject to the
limitation that the knife edges could not allow a
very great tilt ; for it could be rotated about either

in] THE EARTH AS A CLOCK 95

of three axes at right angles. The disc was set
spinning within the circle CC with very great rapidity,
and was then put in position with KK resting on
the Vs. The pull of the* string up and the weight
acting down both passed through the centre of
gravity and did not tend to give any spin to alter
the direction of the axis of rotation of the disc.
Any friction which might come into play as a side-
way force would be so small that it would be very
slow in altering the direction of the axis since the
rate of spin was very great. The direction of the
axis of spin of the disc therefore tended to persist
in the same direction in space. If, for instance, it
was pointed at a particular star, it remained pointing
at that star. Or let us suppose that the axis pointed
at any star on the horizon. As a star begins to rise,
in general it moves partly up and partly along the
horizon, and the latter component of the motion
can be shown to be the same for all stars just rising
to an observer at the same place. Foucault directed
a microscope to a finely divided scale placed across
the outer circle near K, and he found that if the
disc was set spinning and had its axis directed to
any point on the horizon it turned round always in
the direction N.E.S.W., and its motion along the
horizon Avas the same as that of a star rising at the
point ; the rotation having the same rate, in fact, as
that of the plane of his pendulum.

96 THE EARTH [OH.

Foucault also considered what should happen to
a gyroscope in which the axis of spin can only move
round horizontally. We may think, for example, of
the gyroscope in fig. 29 as having the inner circle (7(7
fixed to the outer circle EE so that the axis of the
disc D is horizontal. As the Earth rotates, the
vertical is continually changing its direction in space.
If the gyroscope were not spinning, its centre of
gravity would be pulled at once by the weight into
every new position of the vertical through the point
of suspension ; we should have merely a plumb-bob
and line. But when the disc is spinning very rapidly,
the tendency of the axis to persist in its direction
does not allow this immediate adjustment to take
place, and the centre of gravity is in general not
directly under the point of suspension. Thus the
weight and the pull of the string not being exactly
in one line but in parallel lines, impart a little spin
which, compounded with the spin of the disc, tends
to make the axis move into the N. and S. line. A
special case must serve as illustration, for the com-
plete action is hardly to be followed without mathe-
matical representation. Let the gyroscope be on the
equator, and let the axis of its disc be horizontal and
directed east and west. Fig. 30 (a) is an elevation
as seen from the south, W being the vertical. Let
us suppose that the top part of the disc is moving
southwards, and that the spin is represented by OX,

Ill]

THE EARTH AS A CLOCK

97

fig. 28. Now let the instrument be carried round by
the Earth's rotation to a region fig. 30 (6) where the
direction of the vertical is V V. If the direction of
the axis of spin persisted, the pull of the string and
the weight would have to be as in fig. 30(6), and
they would give a spin round the N. and S. line in
which the top part of the disc would move eastward.
The tendency to persistence in direction of spin

(a)

Fig. 30.

introduces, then, a new spin about an axis at right
angles to the initial spin, and if we represent this
new spin by OA, fig. 28, the resultant is OB, which
is nearer to the meridian. It can be shown that in
whatever direction we have the axis to start with, the
action is of the same kind, tending to bring it nearer
to the meridian. When the meridian is reached,
the motion gained carries the axis through it and,
P. 7

98 THE EARTH [CH.

were there no friction, the axis would go as far on
the other side, would then return and would continue
to vibrate to and fro. But through friction the
vibrations will gradually be lessened and if the spin
of the disc is maintained it will ultimately settle
down pointing true north. Foucault could not main-
tain the spin of his disc sufficiently to verify this,
but he expounded the principle very clearly.

It has lately been carried out in practice in that
very remarkable invention by the brothers Anschiitz,
termed the Gyro Compass1. In this compass the
gyroscope disc is represented by a 3-phase electric
motor, to which the current is fed through the sus-
pension, and the spin of the motor is maintained at
about 300 revolutions per second There is a special
arrangement for damping out the vibrations about
the N. and S. line, and the axis of spin ultimately
settles in that line. The instrument thus constitutes
a mariner's compass, pointing always to the true
north. It is, of course, quite free from the devia-
tions due to iron or steel which the magnetic needle
displays. It appears to act wonderfully well, and if
the expense of its construction could be lessened
it would no doubt entirely displace the magnetic
compass on all large ships.

Both observation and experiment thus confirm the
supposition, already inevitable from a consideration

1 The Anschutz Gyro Compass. Eliott Brothers, London, 1910.

in] THE EARTH AS A CLOCK 99

of forces, that the Earth spins round its axis.
Observation further shows that it spins at a rate so
nearly approaching uniformity that the time of one
revolution does not change by more than a quite
immeasurably small fraction of a second in a year.

We can see how the uniformity is preserved by
considering how spins are made or are changed.
They may be made or changed by the action of
forces not passing through the centre of gravity of
the body spinning, or they may be changed in rate
by alteration of shape of the body.

To illustrate the mode of making or changing
spins by sideway forces, let us suppose that we hang
up a ball by a string and deliver a blow full on it
directed through its centre ; it merely vibrates to and
fro, pendulum- wise. But if a peg projects from one
side and we deliver the blow sideways on the peg,
the ball moves pendulum-wise and at the same time
spins. Thus a spin is made by a sideway force, a
force acting not through but to one side of the
centre of gravity. If while the ball is spinning we
hit it full we may so time the blow and its strength
that we stop the swing, but the spin still persists, for
the force applied was through the centre of gravity.
But if we hit the peg we give a blow at one side
of the centre of gravity. We shall then alter the
spin and may even stop it.

Now the Earth is acted on by forces from outside,

7—2

100

THE EARTH

[CH.

and chiefly from the Sun and Moon. The resultant
attractions of these bodies go almost exactly through
its centre — and at present we will suppose they go
quite exactly through it — and so they are unable to
alter the spin.

We shall see later that we are obliged to suppose
that this is not quite true,
though we have no certain
evidence as yet of altera-
tion of spin.

Again, if a spinning
body changes its shape its
spin may change, and so
if the Earth were, for ex-
ample, decreasing its equa-
torial bulge, its rotation
might be speeding up.
Any bringing together of
the matter of a spinning
system towards the axis
Flg< 31* of rotation makes it go

round the axis in a less time. We may illustrate
this by the apparatus represented in fig. 31, where
A and B are two balls sliding on rods CD and EF,
the two sides of a frame CDFE hung by a string
GH. To A and B are attached strings coming
together at K, and to K is attached the string KL.
Now let the frame and balls be set whirling round

in] THE EARTH AS A OLG€K 101

the axis GHKL. If the string KL is pulled down,
A and B move nearer to the axis and the balls move
round more turns per second. If KL is released A
and B move down and further from the axis, and
the balls move round fewer turns per second.

With the Earth just the same principle holds.
If it were contracting at a sensible rate the spin
would increase and the day would decrease. It is
very probable that the Earth has contracted in the
past, and it may be very slowly contracting now.
But if so, the rate is so slow that any quickening
which it might have produced in the rotation is
probably more than counterbalanced by a slowing
down which the tides have produced in a way ex-
plained hereafter.

In any case the effect is very minute, as certain
ancient records of eclipses show. Eclipses occur in
series at definite intervals, which we can express in
terms of the present time of rotation of the Earth,
and so we can reckon back from eclipses observed
at the present time to eclipses which ought to have
occurred in the past. Some few actual eclipses
recorded by Assyrians, Babylonians, Egyptians, or
Greeks, appear to agree very nearly with such cal-
culated eclipses. Thus there is one series of eclipses
of the Sun at intervals of very nearly 29 years, and
the sum of 18 of these intervals is almost exactly 521
years. There was one of this 521 year series in 1843.

102 THE EARTH [OH.

Reckoning back, a total eclipse shoulcThave occurred
on June 14, B.C. 763, and the path of totality should
have passed 100 miles or more north of Nineveh.
We have a record of a total eclipse at Nineveh about
this time, in all probability that calculated. There
are, however, certain difficulties in making the cal-
culations exactly fit the records of these ancient
eclipses. Thus observation made totality in B.C. 763
at Nineveh ; calculation makes it to the north. But
Mr Cowell has shown that calculations fit the records
much better if the day is lengthening by ^jhr second
per century ! If we assume that this is occurring,
going back a century, the average day during the
century is ffo second shorter than the present day,
and as there are 36,500 days in the century, the
actual century will be shorter than a century made up
of our present days by about 36500/400 = 90 seconds.
Going back 2500 years, nearly arriving at the Assyrian
eclipse, 25 centuries of our days would exceed the
actual 25 centuries by 365 x 2500 x 25/400 seconds or
15 hours. There is no reason to suppose that the
change has been much greater than this. So that
it is probably safe to conclude that the Earth is
rotating so uniformly that it has not lost nearly so
much as a day in 2500 years.

The Earth, then, spins round practically uniformly
under the sky, and the fixed stars appear in con-
sequence to return to the same place at equal

in] THE EARTH AS A CLOCK 103

intervals. But our ordinary day is not exactly one of
these intervals. It is fixed by the interval between
successive returns of the Sun. When, however, we
time the Sun's return to the meridian, to the south
line in the sky, by a very good clock we find that
it does not give us uniform days. Our actual 24
hours' day is only the average interval between
successive passages of the Sun across the meridian.

Let us suppose that we have a uniform clock, a
'Mean Solar Clock,' of which 24 hours is exactly
the average interval between the Sun's successive
passages across the meridian or south line through-
out the year. Then according to the clock the Sun
is sometimes fast, sometimes slow, and for two
reasons which we can examine separately.

The first reason is that the Earth moves round
the Sun in an ellipse, with its greatest speed when
nearest the Sun and its least when farthest away ;
and the second reason is that the Earth's axis is not
perpendicular to the plane of its orbit

Taking the first reason, let fig. 32 represent the
orbit of the Earth, its ellipticity being grossly ex-
aggerated, and, as we are treating the two reasons
separately, let us suppose that the Earth is spinning
round an axis perpendicular to the plane of the orbit.
Let A be the Earth's position when nearest the Sun,
when it is moving fastest, and let P be the point
where the Sun is due south. Next day when the

104

THE EARTH

[CH.

Sun is due south for the same point P let the Earth
have moved to B. We enormously exaggerate the
distance AB in the figure. Then we see that the
Earth has moved more than once round, and by
the angle SBE, where BE is parallel to SA, or by
the angle BSA. Now let us go round to the other
side of the orbit six months later when the Earth

Fig. 32.

is at C and the Sun is due south for the point P.
Next day when it is due south for that point let the
Earth have travelled CD, which is less than AB.
The Earth must have spun more than once round
by the angle SDF, where DF is parallel to SC,
or by the angle DSC, which is obviously less than
BSA. Hence the time between two successive

m] THE EARTH AS A CLOCK 105

southings of the Sun is less when we are at G, that
is about July 1, than when we are at A, that is about
December 31.

Now let us consider the second reason for the
unequal lengths of the solar day, viz. that the
Earth's axis of spin is not perpendicular to the plane
of the Earth's orbit round the Sun. Let us assume
that the last investigation has shown us the effect
of varying speed in an elliptic orbit, and that we
may investigate the present effect while supposing
that the Earth goes round the Sun at uniform speed
in a circle. We can see at once that if the axis were
perpendicular to the plane of the orbit then the
times between successive southings of the Sun would
always be exactly equal for a given point on the
Earth's surface, and for every point day and night
would each be 12 hours. As the Earth went round
the Sun, the Sun would appear to go round the Earth
in the equator of the sky, half way between the sky
poles. But the Earth's axis points about 23° away
from the perpendicular to the plane of the Earth's
orbit or the plane of the ecliptic. The Sun, there-
fore, in its yearly course round the sky does not
appear to move round the equator but in the
circle in which the plane of the orbit meets the

Let the Earth spin counter-clockwise round its
axis PP (fig. 33) at the centre of the equatorial sky

106

THE EARTH

[CH.

circle EBQA, and let ACBD be the circle in which
the plane of the orbit cuts the sky — the ecliptic.

The relative positions of Earth and Sun will be
just the same if we keep the Earth at the centre and
make the Sun move uniformly and counter-clockwise
round the circle ACBD or the ecliptic. Twice in

p

Fig. 33.

the year — on March 21 at A and on September 23
at B — the Sun is on the equator, and day and night
are equal all over the Earth.

Now let us set an imaginary sun — we will call
it the mean Sun — to travel at uniform speed round
the sky equator AEBQ and so that it goes through
A and again through B, with the real Sun moving

in] THE EARTH AS A CLOCK 107

round the ecliptic ACBD. Then if this imaginary
sun could replace the real Sun evidently day
and night would always be equal, and if we con-
structed a perfect clock to show 12 noon for two
consecutive passages across the meridian of the
mean Sun it would show 12 noon for every passage.
Such a clock would be said to keep Mean Solar
Time.

If PorSm is the meridian of a certain place on
the Earth's surface, this meridian must be supposed
to sweep round the sky from left to right as seen
from the outside. When it meets the mean Sun Sm
then for that place the time is mean noon.

Now starting the mean Sun Sm and the true Sun
8 from A on March 21, they move from left to right
as seen from outside, and a short time later when
one is at 8m the other is at 8, where A8 = A8m. It
is evident from the figure that for a short time A8
is less than ACT. This means that the meridian as it
moves from left to right as looked at in the figure
meets 8 before it meets Sm. Then the true Sun is
before the clock. But by June 21, when the mean
Sun is at E the true Sun is at (7, and the meridian
again meets them at the same instant : thus between
March and June there is a time, which is early in
May, when the true Sun is a maximum amount in
advance of the clock. As we have seen, on Septem-
ber 23 the two suns coincide, but a little time before

108

THE EARTH

[CH.

that it is easy to see that the meridian meets the
mean Sun first or the true Sun is behind the clock.
Similarly it can be seen that between September and
January it is in front, and between January and
March it is behind.

Thus we have two effects, that due to ellipticity,

10-

6-

5-

10-

10

5H

0

5-
10-
15 J

< 5

< <0 \0

Fig. 34.

which makes the Sun before the clock half the year
and behind the other half, and that due to inclination
of the axis, which makes the Sun before in one
quarter and behind in the next. In fig. 34 a we re-
present these two effects separately, the continuous
line being the ellipticity effect, and the dotted line
the inclination effect. In fig. 34 b we represent the

in] THE EARTH AS A CLOCK 109

sum of the effects. These figures are taken from
Godfray's Astronomy.

The nett amount by which the true Sun is behind
or in front of the mean Sun or the perfect clock
is called the * equation of time,' and it is reckoned
positive when the Sun is slow, negative when the
Sun is fast.

A sundial keeps true solar time, and so the
1 equation of time ' has to be added to its indication
to give the clock time. When the equation of time
is positive the sundial is slow ; also when it is positive
sunrise is nearer clock noon than is sunset, and this
is very noticeable in January.

The Sun, then, though in the long run he rules
the length of the day, does not keep regular time
day by day as tested by a uniform clock, and so he
fails us as a regulator. How are we to test whether
our clock is uniform ?

For this purpose we must use the fixed stars,
which come round night after night to the south
meridian at very nearly equal intervals. They are
watched and the clocks are rated by them at
Greenwich and at other observatories. But the
hand of the sky clock is not the line to any one
star, nor even to a point fixed relatively to the
stars. It is the line to the point A in fig. 33, the
1 first point of Aries,' and this point travels slowly
round the equator, completing its circle in 25,800

110 THE EARTH [CH.

years. This is due to the fact that the Earth's axis
is not in a fixed direction but is moving round the
perpendicular to the orbit, and A in fig. 33 is moving
round DACB. The Sun and Moon act on the equa-
torial bulge of the Earth in such a way as to make
the Earth wobble or precess, and the wobble, ac-
complished in 25,800 years, is superposed on the spin.
Through it the whole sky slowly rolls round the
perpendicular to the orbit, and that is why any
particular star is unsuitable for a perfect time-
keeper. The time between the successive passages
of the first point of Aries across the meridian is

23 hours 56 minutes 4*09 seconds, and a ' sidereal'
clock which is perfectly rated by the stars should show

24 o'clock every time the first point is due south.
To get the time of passage of the first point, an

ever shifting point, from observations on so-called
fixed stars, calculations are necessary, depending on
the position of the star observed. In practice the
calculations are turned the other way about, so as to
give the times of passage of certain stars across the
meridian after the passage of the first point of Aries.
These times are set forth in the Nautical Almanac,
and the stars are called ' Clock Stars.'

In making the determinations two other disturbing
effects must be taken into account. Over and above
the effect in precession the moon produces another
very slight wobble in the direction of the Earth's

Ill]

THE EARTH AS A CLOCK

111

axis, which is accomplished in 19 years so that not
only does the sky roll slowly round but it quivers
as it rolls. This 19 years' quiver is called 'nutation.'

The other effect is due to the finite speed of travel
of light and it is termed ' aberration.'

For our present purpose we may describe aber-
ration as the alteration in the direction in which
something appears to come to us as we change the
direction or the speed of our own motion.

O Speed of Wind

s D

Fig. 35.

We have an excellent example of aberration in
the direction of the wind on a steamer. Suppose
that for a man on a steamer at rest the wind is from
the west. If the steamer is travelling due north,
then to an observer on the steamer the wind will
appear to be coming from somewhere between N.
and W. The motion of the air relative to the steamer

112

THE EARTH

[CH.

may be obtained by drawing from a point O (fig. 35)
one line OW representing the velocity of the wind
relative to the sea, and another line OS representing
the reverse of the velocity of the steamer, really the
velocity of the sea relative to the steamer. Drawing
a parallelogram on these two lines its diagonal OD
represents the direction of the wind relative to the

D c

(a) (b)

Fig. 36.

observer on the steamer. The change in the direction
of the wind from OW to OD is its aberration.

A similar aberration of wind is a common ex-
perience of every cyclist. His motion turns a wind
coming from any point in the front into one more
nearly a head wind and a wind from behind is
lessened in its effective speed, or may even be turned
into a head wind.

In fig. 36 a or & let OW represent the velocity

in] THE EARTH AS A CLOCK 113

and direction of the wind over the surface of the
ground, OC the velocity of the ground towards the
cyclist, that is the reverse of the velocity of the
cyclist over the ground ; then OZ>, the diagonal of
the parallelogram on OW and OC represents the
wind as experienced by the cyclist. In fig. 36 a we
see how the wind is made much more of a head
wind and much stronger. In fig. 36 b a wind partly
from behind becomes one with a small component
from the front.

There is a precisely similar effect, however it may
be produced, in the case of light. Just as in the
cases we have considered, our motion as we are
carried round the Sun alters the apparent direction
of the light which we receive from any star. We
have to compound with the velocity of the light the
velocity of the earth reversed and the resultant of
these gives us the direction in which the light ap-
pears to come to us.

Let ABGD (fig. 37) represent the orbit of the
Earth as seen from the north side, the Earth going
round counterclockwise. Consider the light coming
from a star a very long way to the right in the line
CA.

Let the velocity of light be V and that of the
Earth be v.

Drawing parallelograms at A and C with sides V
and reversed v, we get the diagonals SO and S'O'

P. 8

114

THE EARTH

[CH.

as the directions in which the star is seen. At B, V
and - v are opposed, but the direction is not altered,
while at D they are added to each other, and again
without alteration of direction. At an intermediate
point such as K, the effect is intermediate between
that at C and D. Thus the star appears to shift to

Fig. 37.

and fro in the course of the year. The speed of the
Earth v is only about jo<bi7 ^ne speed of light "P>
so that in fig. 37 the change of direction is enor-
mously exaggerated. In fact the star as seen from
A and from C will only be about 20 seconds of arc on
either side of its position as seen from B or from Z>,

in] THE EARTH AS A CLOCK 115

the amount subtended by the swing of a pendulum
which moves one foot to either side of its lowest point
when seen from a distance of two miles. It will be
seen from fig. 37 that in the six months when we are
nearest to the star it is deviated to the left, or the
east, and so passes the meridian a little late. During
the other six months it is to the right or the west,
and so it passes the meridian a little early. For
stars in other parts of the sky the effect is a little less
easy to explain and we shall be content to state that
they appear to move in ellipses always with maximum
excursion of 20 seconds on each side of the mean
position.

The aberration is not only important as giving us
a correction to the times of passage of stars across
the meridian but also as giving us an excellent
method of determining our distance from the Sun.
It has been most carefully measured and its amount
— the 20 seconds or thereabouts of its excursion —
is known probably with great accuracy. This 20
seconds, as may be seen from fig. 37, is equal to v/V
or the ratio of the velocity of the Earth round the
Sun to the velocity of light. But experiments have
been made which have determined the time taken by
light to traverse measured distances on the Earth's
surface, i.e. which have determined V. Hence aber-
ration gives us the velocity v of the Earth in its
orbit. We can therefore find the distance it travels

8—2

116 THE EARTH [CH.

in a year, or the length of its orbit, and thence the
radius.

The sky clock, when we allow for the roll of pre-
cession, the quiver of nutation, and the still smaller
quiver of aberration, keeps time so perfectly that no
terrestrial clock can detect any variation in its rate.
But we must not depend on any one particular star.
For the stars are undoubtedly moving, relative to our
solar system, with velocities usually of the order of
10 to 50 miles a second and, in a few cases, with far
higher velocities. In the nearer stars the components
of these velocities perpendicular to our line of sight
produce displacements which become visible in the
course of years, and a star will gradually gain or
gradually lose on the perfectly rated star clock owing
to its 'proper motion' across the sky. But even in
the nearest star this effect is very minute in the
course of a year and absolutely unmeasurable in a
single day. In the more distant stars it will not be
detected even in the course of ages.

Some of the figures in our dial then are changing
their positions. But we can detect these changes in
course of years by watching the changes of pattern
relative to the background of far more distant stars,
and allow for them.

We are justified, then, in concluding that, as far
as any present records go, the Earth spins practically
at a uniform rate beneath the sky.

in] THE EARTH AS A CLOCK 117

Has the Earth been always spinning, and will it
continue spinning, at the same rate ?

This question admits of an answer definitely in
the negative.

We are certain that there is a slowing down of
the spin due to the tides raised in the ocean by the
Moon and Sun, even though it has been so infini-
tesimal during any time in which we have records
available to show it, that we cannot be sure that it
has amounted to a measurable quantity. The in-
vestigation of this slowing down we owe chiefly to
Sir George Darwin.

To understand how it occurs, we must examine
how the tides are formed, and how they follow the
Moon and Sun as gigantic waves round the Earth.

The combined action of Sun and Moon, the
irregular configuration of the oceans, their varying
depths, and the variations of the tidal effect with
latitude, all conspire to make the actual tides ex-
ceedingly complicated. We shall therefore idealise,
and selecting only the part of the tide due to the
Moon we shall suppose that the Earth and Moon
move round each other in the Earth's equatorial
plane and that the ocean forms a continuous canal
round the equator of uniform depth, not more, say,
than three or four miles.

Though we usually speak of the Moon as going
round the Earth, in reality the two form a doublet

118 THE EARTH [OH.

revolving about their common centre of gravity, each
in its own circle, in a month of 27J days, and it is this
common centre of gravity which pursues, as it were, a
smooth elliptic orbit round the Sun. The Earth and
Moon swing now to one side, now to the other, of the
orbit, but they are, of course, always on opposite
sides of the centre of gravity round which they
swing. We shall leave out of account the forward
motion in the orbit and merely consider that which
alone concerns us here, the monthly revolutions
round the common centre. As the Earth's mass is
about 80 times that of the Moon, the common centre
of gravity is about 80 times nearer to the centre of
the Earth than it is to the centre of the Moon, i.e. the
two distances are about 3000 miles and 240,000 miles.
Thus the common centre of gravity is about 1000 miles
within the surface of the Earth.

We can perhaps see how the mutual pulls of the
Earth and Moon suffice to guide them in their re-
spective circles by the following consideration. A
proposition in Mechanics tells us that the motion of
the centre of gravity of any system is the same as if
its whole mass were collected there, and all the forces
acting on the system were transferred there un-
changed in magnitude and in direction. The centre
of gravity of the Earth is at its centre, and, as
Newton proved, the forces on it due to the Moon
are equivalent to the single force which the Moon

in] THE EARTH AS A CLOCK 119

collected at its centre would exert on the Earth
collected at its centre. So the centre of the Earth
moves as if the Earth's mass were all collected there
and pulled by the Moon with a force which is pro-
portional to

Earth's Mass x Moon's Mass
Square of distance between their centres *

Similarly the Moon's centre moves as if all its mass
were collected there and as if it were subjected to a
pull equal and opposite to the above.

As the mass of the Earth is 80 times that of the
Moon and as the pulls on the two are equal, each
pound of the Earth is subjected to ^j of the pull to
which each pound of the Moon is subjected. The
guiding force per pound being -fa, the circle which
it describes will only have a radius •£$ of the radius
of the Moon's circle. That is, the equal and opposite
pulls just account for the two bodies going in circles
round the common centre of gravity somewhat as
represented in fig. 38, though that figure is not
drawn to true scale. EE' is the Earth; C is its
centre ; CO' is the circle which the centre describes
in 27i days round the common centre of gravity G ;
M is the Moon and MM' an arc of its circle.

The Moon-pull is not the same throughout the
Earth. It is only on the average the same as at the
centre: being greater on the near, and less on the

120

THE EARTH

[CH.

far parts. This inequality leads to a deformation of
the Earth. Deformation occurs, no doubt, in the
solid body, which is to some extent yielding and not
perfectly rigid, but it is more conspicuous in the
ocean, loose on its surface, and therefore more easily
yielding than the solid earth.

Fig. 38.

In considering the effect of the varying Moon-
pull, let us suppose, in the first place, that the Earth
presents always the same face to the Moon, just as
the Moon presents always the same face to the
Earth, so that, while it revolves round the common
centre of gravity once in 27J days it rotates also
about its axis in the same time.

Let fig. 39 represent an equatorial section of the
Earth, C its centre, G the common centre of gravity

in] THE EARTH AS A CLOCK 121

of Earth and Moon, and P any particle of matter in
the equatorial canal.

The pull required to guide a body in a circle so
that it shall get round in a certain fixed time is
proportional to the radius of the circle. If, then, we
represent the pull on each pound at C by the radius
<7(? of its circle, the pull on each pound at P is repre-
sented by the radius PG of its circle. If we draw the
parallelogram CPMG the pull PG may be resolved
into the two pulls PM and PC, and we can at once

Moon,

Fig. 39.

see the significance of each. The former PM= CG is
the same in magnitude and direction for every particle
of given mass in the canal and is the force needed to
guide it in a circle of radius PM equal to GC. But
if we have revolution, without rotation, round G —
the kind of motion exemplified by a bicycle pedal
when the cyclist keeps his foot horizontal — every
particle does go round in a circle, and the radius of
every circle has the same value PM or CG. Thus

122 THE EARTH [CH.

while C (fig. 39) moves to C' round G, P moves to P'
round M, where C'P' is parallel to OP. So that the
PM component is what is needed for revolution
without rotation. The pull represented by PC is
that needed to guide the matter at P in a circle
with radius PC in a time of 27J days ; that needed
for the rotation apart from the revolution. It has
nothing to do with the Moon. It is supplied by
the Earth's attraction. Or in other words, some of
the weight is used up in guiding the mass at P in
its rotation circle. The consequence of this virtual
reduction of weight is, as we have seen in Chapter I,
a tendency to an equatorial bulge all round. If the
Earth's speed of rotation is increased one effect is an
increase in this component and an increased equa-
torial bulge.

As far then as the Moon's action goes, we need
only regard the component PM=CG and consider
how far the Moon supplies this pull. In fig. 40, let
A' A be the equatorial diameter passing through the
Moon and BB ' the equatorial diameter at right
angles. On the hemisphere facing the Moon its actual
attraction on every particle is greater than it would
be on the same particle at C except near B and
B', i.e. it is greater than the attraction represented
by CG. There is thus an excess over what is needed
to keep the surface matter in its circle of radius PJff,
and the excess gradually increases from B where it

Ill]

THE EARTH AS A CLOCK

123

is practically zero, to A where it is a maximum.
Similarly it increases from B' to A. On the hemi-
sphere away from the Moon the attraction is less
than at (7, so that we may represent it by a pull

Fig. 40.

^

Fig. 41.

towards the Moon equal to <76r, viz. that required to
keep the surface matter in its circle, combined with
a small extra pull in the opposite direction, and as
the attraction at A' is less than that at C by very

124 THE EARTH [CH.

nearly as much as that at A is greater, the extra
pulls away from the surface on the two sides at
equal distances from A A' are very nearly equal, and
we get extra forces over and above those needed for
guidance in the circle, somewhat as represented in
fig. 40.

Now to find the effect of these on the water in
the canal we must resolve each along the vertical
and horizontal. The vertical component merely
diminishes the weight slightly and may be neglected.
The horizontal is the important component and it
will be seen that this horizontal component vanishes
at B and B' and again at A and A', and is a
maximum at the ends of diameters making about 45°
with AA', somewhat as represented in fig. 41. These
horizontal forces would move the water away from B
and B' and heap it up at A and A ', the ends of the
diameter of the Earth pointing to the Moon, as re-
presented with enormous exaggeration in fig. 42, if
the Earth always presented the same face to the
Moon. Though the Moon has no ocean the similar
action of the Earth upon it undoubtedly deforms its
solid body thus, and as it always presents the same
face to the Earth it bulges slightly towards the
Earth and bulges away from it on the opposite side.

Now consider the effect of increasing the rotation
of the Earth from once in 27£ days to once in 24
hours. The Moon heaps up the water at two ends

in] THE EARTH AS A CLOCK 125

Moon

Fig. 42.

Moon

Fig. 43.

Moon\

Fig. 44.

126 THE EARTH [OH.

of a diameter, but the Earth is moving rapidly from
west to east under the heaps. Relative to the
surface of the Earth the heaps move from east to
west. In the canal, then, we have two heaps and
two hollows always travelling from east to west,
two waves, each of length from crest to crest half
the Earth's circumference. They move once round
the Earth while a point on its surface travels from
one position under the Moon to its next position
under the Moon, i.e. in about 25 hours, and they bring
to each point two high tides per journey round. As
the circumference is 25,000 miles this implies a speed
of about 1000 miles an hour.

But now comers in a curious consequence of the
fact that the tides are waves. In a canal, or indeed
in any sheet of water, waves once made and then
allowed to travel on naturally, under the forces called
into play merely by the shapes of the waves, have a
definite speed of travel depending, if they are very
long waves, on the depth of the canal. Each tidal
wave is here 12,500 miles long, a very great length
compared with the depth of the canal, which we
have supposed three or four miles deep. Such a
wave would require a canal 13 or 14 miles deep to
have a speed of 1000 miles an hour under its own
natural forces only. In a four-mile-deep canal the
speed would be only about 550 miles per hour. The
tidal wave, then, going round once in 25 hours and

in] THE EARTH AS A CLOCK 127

having a speed about 1000 miles per hour, is travel-
ling much more rapidly than a natural wave. To
get this greater speed the wave must so arrange
itself on the canal that the Moon-forces shall con-
spire with the natural wave-forces to increase the
speed of travel. The natural forces are always pres-
sures from the crests towards the troughs of the
waves, and if the crests of the tidal waves are at B
and Bf (fig. 43) and the troughs at A and A' instead
of the reverse, it is seen from fig. 41 that the Moon-
forces agree in direction with the natural forces, and
so hurry the motion of every particle and increase
the wave speed. The tide, then, tends to have its
high water at B and B' &s in fig. 43 and to be of
such height that the two sets of forces give it just
the right speed of 1000 miles per hour.

This tendency to have high water just J way
round the Earth from where we might at-first expect
it, is called the inversion of the tide. Were there no
friction the inversion of tide in the canal we are
imagining might be exact, i.e. low water might be
directly under the Moon and on the opposite side
and high water at right angles.

But friction acts in such a way that the Earth,
turning counterclockwise as seen from the north,
leaves high water and low water rather behind,
rather to the west of the places where we might
expect them, somewhat as in fig. 44, and we may

128 THE EARTH [OH.

perhaps give some explanation of this lag as follows,
though the explanation is not quite accurate nor is
it complete.

In a water wave the water has a to and fro motion
as well as an up and down motion and about the
crest of the wave the forward motion is a maximum

Moon

and it helps to transfer the crest to the next point
in advance. About the middle of the trough there
is the maximum backwards motion which helps to
transfer the hollow forward. The frictional resisting
force called into play by this horizontal motion is
a force on the water, backwards at the crest, and
forwards at the trough. We have to consider, then,

Ill]

THE EARTH AS A CLOCK

129

not only the Moon-forces, but also these frictional
forces, and the two sets together must conspire with
the natural forces to hurry the motion in the waves.
Remembering that the high water is somewhere near
B and B' (fig. 45), the horizontal friction-forces on
the water will be somewhat as indicated by the
arrows inside the circle, while the Moon-forces are
those outside ; the friction-forces being much the

Moon Jorct

FrttUvn jorte.

ResulloKl -force

Fig. 46.

smaller set. The effect of their addition to the
Moon-forces is to carry the points of zero horizontal
force round from BB' towards the west, i.e. a little
way round in the clockwise direction and every part
of the force scheme will be similarly carried round.
This can be better seen, perhaps, if we represent
different points on the circumference by points on a
straight line, and the two sets of forces by curves
(fig. 46) where the curve representing the friction-
force should really be a little more to the right. In
p. 9

130 THE EARTH [CH.

order that the compound of friction and Moon-forces
may agree with the internal forces, vanishing at the
same points and having maxima and minima at the
same points, the waves must also be turned round a
little in the clockwise direction, i.e. a little to the west.
The friction between the ocean and the Earth
cannot of itself alter the spin of the Earth as a
whole for the mutual action of two parts of a system
cannot alter the sum total of their angular momenta.
The friction acts indirectly by shifting the positions

Moon,

Fig. 47.

of high and low water, thereby altering the Moon's
pulls and enabling them to put a brake on the Earth.
This will be seen in a general way from tig. 47. The
Moon-pull on the bulge at H is greater than that on
the bulge at H', since the latter is more distant ; the
former tends to lessen the spin, the latter to increase
it, and the net result is a diminution in spin.

To consider the effect a little more exactly, let

Ill]

THE EARTH AS A CLOCK

131

us continue the lines of the two forces at H and H '
to the Moon's centre M (fig. 48). Only if the two
forces were in the proportion of HM to H'M would
their resultant act through G the centre of the Earth,
half-way between H and H'. But if the force along
HM is represented by HM that along H'M must
be represented by a smaller length H"M, and the
resultant is along C'M, where C' is half-way between
H and H". Now CO' is easily seen to be parallel

C1 < •*- 'M

Fig. 49.

to H'H" so that G' is necessarily above CM. Then
the resultant action of the Moon on the two tidal
heaps is a force not through the centre of the
Earth, but on the H side of the centre. It therefore
is what we have termed a sideway force, and it is
always acting to slow down the rate of rotation. So
we conclude that the tides are gradually reducing the
spin of the Earth. After a time the Earth will move

9—2

132 THE EARTH [CH.

so slowly that the tide will no longer be inverted
but it can be shown that it is then displaced by
friction in such a direction that the action still
reduces the Earth's spin.

Let us now turn to the Moon. In the first place
the action of the Earth in raising tides in the Moon
explains at once how she now turns always the
same face towards us, or rotates on her axis once
a month. When she was perhaps much hotter and
perhaps more plastic and certainly younger, the
Earth must have raised very considerable tides in
the solid body as well as in her oceans, if ever she
had oceans. On these the Earth would act as the
Moon acts now on the Earth tides, but much more
considerably. The resultant action would be a force
not through her centre, but a 'sideway' force op-
posing her spin round her axis ; acting in fact as
a brake until the spin was reduced so far that brake
and wheel went round together, the Moon's period
of rotation coinciding with the month. The tides
on the Moon, tides in the slightly plastic body, are
always now at the same parts of her surface, directly
facing and directly opposite to the Earth.

In the second place there is a reaction of the
Earth's tides on the Moon equal and opposite to the
action of the Moon on the Earth's tides. We said
that this was a force on the Earth along C'M (fig. 48),
so that the equal and opposite force on the Moon

in] THE EARTH AS A CLOCK 133

is along M C' not quite directed to the Earth's centre.
Resolving MC' (fig. 49) into MC and M. Z>, the former
being through the Earth's centre, the latter is a small
component in the direction in which the Moon is
moving in her orbit. This force is continually doing
work on the Moon, tending to increase her velocity.
But instead of this tendency being fulfilled there is
an opposite effect. Inasmuch as without this pull
along the path the Moon would be guided along a
circle by the pull towards the Earth's centre; with
the pull she moves slightly outside the circle, moves
in fact in a slowly widening spiral, getting further
and further from the Earth. As she gets further
out, in increasing her distance against the pull to the
Earth's centre she uses up not only all the energy
put in by the pull along her path but also some of
her own kinetic energy ; somewhat as a cyclist going
up a hill slackens speed, because the potential energy
required is more than the energy which he puts in at
the pedals, and so there is a call on and a diminution
in his kinetic energy.

We conclude that in our idealised Earth with an
equatorial canal, the action of the Moon on the tides
is gradually lengthening the day, while the reaction
of the tides on the Moon is gradually driving her
out and lengthening the month.

On the real Earth, with its complicated distribution
of oceans, the action is the same in kind but too

134 THE EARTH [OH.

complicated to allow calculation of the rate at which
the action is going on. But there is a general principle
which enables us to say what is the relation between
the day and the month at any stage. This is the
principle of the Conservation of Angular Momentum
which asserts that in a system no interaction between
the various parts will change its total spin. That
spin is to be estimated by multiplying each pound of
matter in the system by its distance from the axis
round which the spin is to be calculated and by the
component of the velocity perpendicular to the line
drawn to the axis and adding up for the whole
system. In the case of the Earth and Moon the
spin is shared between the Earth and Moon. The
Earth's share is gradually decreasing as the day
lengthens. The Moon's share is gradually increasing,
for her increasing distance more than makes up for
her decreasing velocity, but the sum total for Earth
and Moon is constant. Sir George Darwin has shown
that the slowing down of the Earth's rotation will
continue till the day is 55 of our present days. The
month will then also be lengthened out to 55 of our
present days and the Moon will be more than half
as far again away from us as now. The Moon and
the Earth will be always turning the same face
towards each other, the tides will be at the same
parts of the surface of each, and the tidal brake will
cease to act.

in] THE EARTH AS A CLOCK 135

So far we have left out of account the tides
which the Sun raises. But these are by no means
negligible. Every fortnight, when the Sun and Moon
are in the same or in opposite parts of the sky, we
have spring tides with high water much higher and
low water much lower than at the times half-way
between, when the Sun and Moon are at right-angles
and we have the much smaller neap tides. The
Moon tide is to the Sun tide about as 9 to 4, so that
if the rise is represented by 13 when they are together,
when they are opposite it is represented by about 5.
The solar tides being so appreciable, the action of
the Sun on these tides must also be appreciable
and must tend to reduce the spin of the Earth.
But the reduction is at a very much less rate than
that due to the Moon. It will become more impor-
tant in its effect when the Earth and Moon have come
to an equal day and month which, as we have seen,
works out at 55 of our days. For then the solar
tides will slacken down the spin of the Earth still
further without changing the length of the month
and the Moon tides will again travel round the Earth.
But now, relative to the surface of the Earth, they
will travel in the opposite direction. The action
between the Moon and tides will therefore be
reversed, and the Moon will be gradually drawn
inwards.

Let us now return from this vastly distant future

136 THE EARTH [CH.

to the present day, and then travel back into the
past. The process now going on implies that if we
travelled back we should find the day and month
both shorter and shorter and the Moon nearer and
nearer to the Earth. And when the Moon was nearer
the tides would be higher and the action greater.
At last we should arrive at a time when, as calcu-
lation shows, both day and month were only from
three to five of our present hours, and when the
Moon must have been close in to the Earth. Here
precise calculation ends.

If we make the very probable guess that before
this the Moon and the Earth formed one body we
can go one step further back in the history.

A planet of the joint mass of Earth and Moon,
and of volume somewhat larger than the sum of the
present volumes, as it probably would be, spinning
round in about three hours would be very nearly
unstable ; the weight of the surface parts would only
just suffice to hold them on to the surface. And we
can assign a very probable reason for the small
stability passing over into instability and disruption.
If we could imagine a liquid globe to receive a
deformation — to be pressed in, for instance, at the
ends of a diameter and to be bulged out at the ends
of a perpendicular diameter — and to be then released
it would vibrate somewhat as a bell vibrates, and
in a time depending on its density. That time of

in] THE EARTH AS A CLOCK 137

vibration works out for the liquid globe we have sup-
posed at about 1£ hours. Let us suppose that for the
actual globe it was something of this order, say it
was two hours. The Sun would be raising tides in
the globe, two tides in each day. And through
friction these would be gradually lengthening out
the day. A time might come when the solar tides,
following each other at half-day intervals, would just
agree in period with the period of free vibration.
We should then have 'resonance' and the tides
would become greater and greater until the crest
of one of the waves — perhaps both crests, were
thrown off to form ultimately a separate globe, the
Moon.

There does not seem to be any wild conjecture in
summing up the past and future history of the Earth
and Moon system as follows. Immensely far back in
the past there was a globe revolving round the Sun and
spinning round with a day of very few of our hours.
The Sun raised tides which gradually lengthened the
day until their half-day period just coincided with
the period of natural pulsation of the globe and the
Sun-tides grew so high through this coincidence that
the crests flew off and the Moon was born. At first
day and month coincided, each being perhaps four of
our hours. But the Moon raised tides and her action
on these lengthened the day far more rapidly than
the Sun could, while the reaction of the tides on the

138 THE EARTH [CH. m

Moon drove her ever further away. Meanwhile the
Earth raised tides on the Moon which slowed down
her spin until at length lunar day and month were
the same, as they are now. In the past when Earth
and Moon were nearer to each other the tidal action
must have been more rapid than now, when it is so
slow that even in 2500 years it is only doubtfully
detected.

In the future it will be still slower. But we can-
not doubt that it will continue till the Moon is half
as far away again as now, till the month is twice as
long as now, and the day is as long as the month, so
that the Earth and Moon will present continually
the same face to each other. But the history does
not end there. The smaller solar action will continue
to lengthen out the day without affecting the month
directly and the lunar tides will then travel round
in the opposite direction relative to the surface of
the Earth. The reaction on the Moon will be re-
versed and she will gradually begin to retrace her
spiral, this time towards the Earth, and perhaps at
some enormously distant future time end her journey
by reunion with the parent globe.

INDEX

Aberration, 111, 115
a Centauri, distance of, 36
Air in cyclones, motion of, 90
Airy, 60
Anchor ring, 12
Angular Momentum, Conserva-
tion of, 134
Anschiitz, 98
Antipodes, 8

Aries, first point of, 109, 110
Assyrians, Eclipse records, 101
Attracted spheres, 64, 69
Attracting spheres, 64, 69
Attraction, defect in, 60
Axis, direction of gyroscope, 95

Babylonians, Eclipse records,

101

Balance, common, 72—77
Base-line, 27 ; measuring of, 21
' Base-line ' method, 17, 34 ;

principle of, 18, 20
Bouguer, 53, 55, 57, 60
Boys, Prof. C. V., 68, 69, 71
British Isles, triangulation of,

28

Cassini, 30, 31

Cavendish, 56, 59, 62, 63, 67,

69
Chimborazo, 53, 55, 57

Chronometer, 42

Clarke, Col., 32

'Clock Stars,' 110

Colby, Colonel, 22

Columbus, 2, 3

Compound measuring rod, 23

Condamine, De la, 31, 54

Continental survey, connection

with, 28
Coulomb, 62
Coventry, 5, 13
Cowell, 102
Cross-wires, 15
Cyclone, 88—90
Cylinder, 12

Darwin, Sir George, 117, 134
Day, lengthening of, 133, 136
Deformation of Earth, 120
'Double-suspension' mirror, 78,

79
Dover, 27

Earth, a round globe, 13 ; action
of, in raising tides on Moon,
132; average density of, 47;
motion of axis of, 110 ; curva-
ture of, 58 ; equatorial section,
120; mass of, 37, 46; mean
density of, 47 ; orbit of, 103 ;
rotation of, 88 ; spin of, slow-

THE

CAMBRIDGE MANUALS
OF SCIENCE AND LITERATURE

Published by the Cambridge University Press

GENERAL EDITORS

P. GILES, Litt.D.

Master of Emmanuel College

and
A. C. SEWARD, M.A., F.R.S.

Professor of Botany in the University of Cambridge

SIXTY VOLUMES NOW READY
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Ancient Assyria. By Rev. C. H. W. Johns, Litt.D.

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A History of Civilization in Palestine. By Prof. R. A. S.

Macalister, M.A., F.S.A.

China and the Manchus. By Prof. H. A. Giles, LL.D.
The Civilization of Ancient Mexico. By Lewis Spence.
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and J. Logan Stout, LL.B. (N.Z.).
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Hamilton Thompson, M.A., F.S.A.
The Historical Growth of the English Parish Church. By A.

Hamilton Thompson, M.A., F.S.A.
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Ancient Stained and Painted Glass. By F. S. Eden.

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The Early Religious Poetry of the Hebrews. By the Rev.

E. G. King, D.D.
The Early Religious Poetry of Persia. By the Rev. Prof. J.

Hope Moulton, D.D., D.Theol. (Berlin).

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The History of the English Bible. By the Rev. John Brown,

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English Dialects from the Eighth Century to the Present Day.

By W. W. Skeat, Litt.D., D.C.L., F.B.A.
King Arthur in History and Legend. By Prof. W. Lewis

Jones, M.A.

The Icelandic Sagas. By W. A. Craigie, LL.D.
Greek Tragedy. By J. T. Sheppard, M.A.
The Ballad in Literature. By T. F. Henderson.
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The Idea of God in Early Religions. By Dr F. B. Jevons.

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Cash and Credit. By D. A. Barker, I.C.S.

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and V/ales). By G. Glover Alexander, M.A., LL.M.

BIOLOGY

The Coming of Evolution. By Prof. J. W. Judd, C.B., F.R.S.
Heredity in the Light of Recent Research. By L. Doncaster,

M.A.

Primitive Animals. By Geoffrey Smith, M.A.
The Individual in the Animal Kingdom. By J. S. Huxley, B.A.
Life in the Sea. By James Johnstone, B.Sc.
The Migration of Birds. By T. A. Coward.
Spiders. By C. Warburton, M.A.
House Flies. By C. G. Hewitt, D.Sc.
Earthworms and their Allies. By F. E. Beddard, F.R.S.

ANTHROPOLOGY

The Wanderings of Peoples. By Dr A. C. Haddon, F.R.S.
Prehistoric Man. By Dr W. L. H. Duckworth.

GEOLOGY

Rocks and their Origins. By Prof. Grenville A. J. Cole.
The Work of Rain and Rivers. By T. G. Bonney, Sc.D.
The Natural History of Coal. By Dr E. A. Newell Arber.
The Natural History of Clay. By Alfred B. Searle.
The Origin of Earthquakes. By C. Davison, Sc.D., F.G.S.

BOTANY

Plant-Animals : a Study in Symbiosis. By Prof. F. W. Keeble.
Plant-Life on Land. By Prof. F. O. Bower, Sc.D., F.R.S.
Links with the Past in the Plant- World. By Prof. A. C. Seward.

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The Earth. By Prof. J. H. Poynting, F.R.S.

The Atmosphere. By A. J. Berry, M.A.

The Physical Basis of Music. By A. Wood, M.A.

PSYCHOLOGY

An Introduction to Experimental Psychology. By Dr C. S.

Myers.
The Psychology of Insanity. By Bernard Hart, M.D.

INDUSTRIAL AND MECHANICAL SCIENCE

The Modern Locomotive. By C. Edgar Allen, A.M.I.Mech.E.

The Modern Warship. By E. L. Attwood.

Aerial Locomotion. By E. H. Harper, M.A., and Allan E.

Ferguson, B.Sc.

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The Aryans. By Prof. M. Winternitz.

The Peoples of India. By J. D. Anderson.

Prehistoric Britain. By L. McL. Mann.

The Balkan Peoples. By J. D. Bourchier.

The Evolution of Japan. By Prof. J. H. Longford.

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The West Indies. By Sir Daniel Morris, K.C.M.G.

The Royal Navy. By John Leyland.

Gypsies. By John Sampson.

English Monasteries. By A. H. Thompson, M.A.

A Grammar of Heraldry. By W. H. St John Hope, Litt.D.

Celtic Art. By Joseph Anderson, LL.D.

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Pantomime. By D. L. Murray.

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Wireless Telegraphy. By C. L. Fortescue, M.A.

Rontgen Rays. By Prof. W. H. Bragg, F.R.S.

BIOLOGY

Bees and Wasps. By O. H. Latter, M.A.

The Life-story of Insects. By Prof. G. H. Carpenter.

The Wanderings of Animals. By H. F. Gadow, M.A., F.R.S.

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UNIVERSITY OF CALIFORNIA LIBRARY